GCSE Maths: Interior and exterior angles of polygons (Higher)
Part of the GCSE Maths angles, polygons and symmetry chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Interior and exterior angles of polygons
A polygon is a closed shape with straight sides. The angle inside the shape at a corner is the interior angle. If you extend one side beyond a corner, the angle between the extension and the next side is the exterior angle. At every corner, the interior angle and exterior angle sit on a straight line, so interior + exterior = 180°.
Sum of the interior angles. Choose one corner of a polygon and join it to every other corner you can. This cuts the polygon into triangles. A quadrilateral makes 2 triangles, a pentagon makes 3, a hexagon makes 4, and in general an n-sided polygon makes n − 2 triangles. Each triangle has an angle sum of 180°, so
sum of interior angles = (n − 2) × 180°
| Polygon | Sides n | Triangles n − 2 | Angle sum |
|---|---|---|---|
| Triangle | 3 | 1 | 180° |
| Quadrilateral | 4 | 2 | 360° |
| Pentagon | 5 | 3 | 540° |
| Hexagon | 6 | 4 | 720° |
| Heptagon | 7 | 5 | 900° |
| Octagon | 8 | 6 | 1080° |
| Nonagon | 9 | 7 | 1260° |
| Decagon | 10 | 8 | 1440° |
Regular polygons. In a regular polygon all the sides and all the angles are equal. So each interior angle is the sum divided by n: (n − 2) × 180° ÷ n. For a regular hexagon this is 720 ÷ 6 = 120°.
Exterior angles. Imagine walking round a polygon. At each corner you turn through the exterior angle. By the time you are back where you started and facing the same way, you have made exactly one full turn. So the exterior angles of any polygon add up to 360°. In a regular polygon they are all equal, so each exterior angle is 360° ÷ n. For a regular hexagon: 360 ÷ 6 = 60°, and the interior angle is 180 − 60 = 120°. This shortcut is often the quickest method.
Key ideas
- Interior angle + exterior angle = 180° at each corner.
- Sum of interior angles = (n − 2) × 180°.
- Sum of exterior angles = 360° (for every polygon).
- Regular polygon: each exterior angle = 360° ÷ n; each interior angle = 180° − exterior angle.
- To find the number of sides of a regular polygon: n = 360 ÷ exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 8 sides, so it is a octagon.n = 8
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.8 − 2 = 6 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.6 × 180 = 1080°
Check. Check with the formula (n − 2) × 180 = (8 − 2) × 180 = 1080°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular nonagon has n = 9 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(9 − 2) × 180 = 7 × 180 = 1260°
- Share the sum equally. In a regular polygon all 9 interior angles are equal, so each one is the sum divided by 9.1260 ÷ 9 = 140°
Check. Another way: each exterior angle is 360 ÷ 9 = 40°, and 180 − 40 = 140°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a pentagon add up to (5 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
- Add the angles we know. We need their total to take from the sum.100 + 120 + 90 + 110 = 420°
- Subtract from 540°. What is left is the unknown angle.540 − 420 = 120°
x = 120°
Check. Add all the angles: 100 + 120 + 90 + 110 + 120 = 540°, which is 540°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 24°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 24°, so the number of them is 360 ÷ 24.360 ÷ 24 = 15 sides
Check. 15 × 24 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 156°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 156 = 24°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 24° angles are equal, so divide 360° by the exterior angle.360 ÷ 24 = 15 sides
Check. Each interior angle of a regular 15-sided polygon is (15 − 2) × 180 ÷ 15 = 156°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Watch out
Do not divide 180° by the number of sides to get the interior angle, and do not mix up sum and each. The formula (n − 2) × 180° gives the sum of all the interior angles; you only divide by n for a regular polygon. Also, the exterior angles always add to 360°, whatever the polygon.Exercise 17G Angles in polygons: sums and missing angles
Non-calculatorKey words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
Check. Check with the formula (n − 2) × 180 = (5 − 2) × 180 = 540°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
Check. Check with the formula (n − 2) × 180 = (6 − 2) × 180 = 720°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 8 sides, so it is a octagon.n = 8
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.8 − 2 = 6 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.6 × 180 = 1080°
Check. Check with the formula (n − 2) × 180 = (8 − 2) × 180 = 1080°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 10 sides, so it is a decagon.n = 10
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.10 − 2 = 8 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.8 × 180 = 1440°
Check. Check with the formula (n − 2) × 180 = (10 − 2) × 180 = 1440°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Split the polygon into triangles from one corner. The number of triangles is n − 2, and each has angle sum 180°.
- Count the sides. The shape has 12 sides, so it is a dodecagon.n = 12
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.12 − 2 = 10 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.10 × 180 = 1800°
Check. Check with the formula (n − 2) × 180 = (12 − 2) × 180 = 1800°; and for a triangle (n = 3) it gives 180°. ✓
Watch out. The number of triangles is two fewer than the number of sides, not equal to it.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a pentagon add up to (5 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
- Add the angles we know. We need their total to take from the sum.100 + 120 + 90 + 110 = 420°
- Subtract from 540°. What is left is the unknown angle.540 − 420 = 120°
x = 120°
Check. Add all the angles: 100 + 120 + 90 + 110 + 120 = 540°, which is 540°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a hexagon add up to (6 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
- Add the angles we know. We need their total to take from the sum.130 + 100 + 115 + 125 + 95 = 565°
- Subtract from 720°. What is left is the unknown angle.720 − 565 = 155°
y = 155°
Check. Add all the angles: 130 + 100 + 115 + 125 + 95 + 155 = 720°, which is 720°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a heptagon add up to (7 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 7 sides, so it is a heptagon.n = 7
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.7 − 2 = 5 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.5 × 180 = 900°
- Add the angles we know. We need their total to take from the sum.150 + 110 + 120 + 140 + 125 + 135 = 780°
- Subtract from 900°. What is left is the unknown angle.900 − 780 = 120°
z = 120°
Check. Add all the angles: 150 + 110 + 120 + 140 + 125 + 135 + 120 = 900°, which is 900°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a pentagon add up to (5 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 5 sides, so it is a pentagon.n = 5
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.5 − 2 = 3 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.3 × 180 = 540°
- Add the angles we know. We need their total to take from the sum.128 + 98 + 112 + 102 = 440°
- Subtract from 540°. What is left is the unknown angle.540 − 440 = 100°
a = 100°
Check. Add all the angles: 128 + 100 + 98 + 112 + 102 = 540°, which is 540°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of a hexagon add up to (6 − 2) × 180°. Subtract the angles we know.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
- Add the angles we know. We need their total to take from the sum.105 + 120 + 135 + 95 + 140 = 595°
- Subtract from 720°. What is left is the unknown angle.720 − 595 = 125°
b = 125°
Check. Add all the angles: 105 + 120 + 135 + 95 + 140 + 125 = 720°, which is 720°. ✓
Watch out. Work out the correct total first: (n − 2) × 180. Do not use 360° (that is for a quadrilateral).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Use the formula (n − 2) × 180 = sum, and work backwards.
- Write the formula as an equation. The interior angles of an n-sided polygon add up to (n − 2) × 180°.(n − 2) × 180 = 1260
- Divide both sides by 180. This undoes the multiplication by 180.n − 2 = 1260 ÷ 180 = 7
- Add 2 to both sides. This undoes the subtraction.n = 7 + 2 = 9
Check. (9 − 2) × 180 = 7 × 180 = 1260°. ✓
Watch out. Remember to add the 2 back after dividing.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular pentagon has n = 5 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(5 − 2) × 180 = 3 × 180 = 540°
- Share the sum equally. In a regular polygon all 5 interior angles are equal, so each one is the sum divided by 5.540 ÷ 5 = 108°
Check. Another way: each exterior angle is 360 ÷ 5 = 72°, and 180 − 72 = 108°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular hexagon has n = 6 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(6 − 2) × 180 = 4 × 180 = 720°
- Share the sum equally. In a regular polygon all 6 interior angles are equal, so each one is the sum divided by 6.720 ÷ 6 = 120°
Check. Another way: each exterior angle is 360 ÷ 6 = 60°, and 180 − 60 = 120°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular octagon has n = 8 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(8 − 2) × 180 = 6 × 180 = 1080°
- Share the sum equally. In a regular polygon all 8 interior angles are equal, so each one is the sum divided by 8.1080 ÷ 8 = 135°
Check. Another way: each exterior angle is 360 ÷ 8 = 45°, and 180 − 45 = 135°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular decagon has n = 10 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(10 − 2) × 180 = 8 × 180 = 1440°
- Share the sum equally. In a regular polygon all 10 interior angles are equal, so each one is the sum divided by 10.1440 ÷ 10 = 144°
Check. Another way: each exterior angle is 360 ÷ 10 = 36°, and 180 − 36 = 144°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular dodecagon has n = 12 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(12 − 2) × 180 = 10 × 180 = 1800°
- Share the sum equally. In a regular polygon all 12 interior angles are equal, so each one is the sum divided by 12.1800 ÷ 12 = 150°
Check. Another way: each exterior angle is 360 ÷ 12 = 30°, and 180 − 30 = 150°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Method. The sum must equal (n − 2) × 180 for a whole number n.
- Write the equation. The interior angles of an n-sided polygon add up to (n − 2) × 180°.(n − 2) × 180 = 900
- Divide by 180. n − 2 = 900 ÷ 180 = 5
- Decide. A whole number of sides (5 + 2 = 7) works, so the answer is yes.n = 7: a heptagon
Check. (7 − 2) × 180 = 900°. ✓
Watch out. A sum like 1000° would be impossible because 1000 ÷ 180 is not a whole number.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the angle sum of a hexagon, then subtract the five known angles.
- Count the sides. The shape has 6 sides, so it is a hexagon.n = 6
- Split the polygon into triangles. Join one corner to every other corner that it is not already joined to. This makes n − 2 triangles, and each triangle has angles that add up to 180°.6 − 2 = 4 triangles
- Multiply by 180°. The angle sum of the polygon is the number of triangles times 180°.4 × 180 = 720°
- Add the five known angles. They are all 130°.5 × 130 = 650°
- Subtract from the sum. 720 − 650 = 70°
Check. 5 × 130 + 70 = 720°. ✓
Watch out. Multiply, do not add five times by hand: it is easy to slip.
Key words. An angle measures a turn and is written in degrees (°). A reason is the angle fact you used, written in words, for example “angles on a straight line add up to 180°”. In an exam you gain a mark for the reason as well as for the number.
Method. Add the expressions and put them equal to 540°. Solve the equation, then substitute to find the angles.
- Choose the angle fact. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.The interior angles of an n-sided polygon add up to (n − 2) × 180°.
- Write an equation. The angles must add up to 540°, so we add all the expressions and put the answer equal to 540.x° + 2x° + (x + 30°) + (2x − 20°) + (3x − 10°) = 540°
- Collect like terms. Add all the x terms together, and add all the plain numbers together.9x = 540
- Divide both sides by 9. 9x means 9 × x, so dividing by 9 leaves x on its own.x = 540 ÷ 9 = 60
- Substitute x = 60 to find each angle (if the question needs them). The question may ask for the size of an angle, not just for x.x° = 1 × 60 = 60°
2x° = 2 × 60 = 120°
x + 30° = 1 × 60 + 30 = 90°
2x − 20° = 2 × 60 − 20 = 100°
3x − 10° = 3 × 60 − 10 = 170°
Check. Add the angles: 60 + 120 + 90 + 100 + 170 = 540°, which is 540°. ✓
Watch out. The value of x is not the size of the angle. Substitute x into each expression to find the angles, and check that they add to the right total.
Method. Split the polygon into triangles from one corner and add up the angles of the triangles.
- Choose one corner. Join it to every corner it is not already next to.This uses n − 3 lines (diagonals)
- Count the triangles. The n − 3 diagonals cut the polygon into one more piece than the number of cuts.n − 3 + 1 = n − 2 triangles
- Add the angles. The angles of the triangles fill the corners of the polygon exactly, and each triangle has angle sum 180°.sum = (n − 2) × 180° ■
Check. For a hexagon (n = 6): 4 triangles, 4 × 180 = 720°. ✓
Watch out. The triangles do not overlap and they fill the whole polygon.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Check how many triangles a hexagon is split into.
- Count the triangles. From one corner of a hexagon you can draw 3 diagonals, which makes 4 triangles (not 6).6 − 2 = 4 triangles
- Spot the mistake. Zoe multiplied by the number of sides instead of by the number of triangles.wrong: 6 × 180
- Correct the answer. 4 × 180 = 720°
Check. A hexagon has angles of 120° if regular: 6 × 120 = 720°. ✓
Exercise 17H Regular polygons and exterior angles
Non-calculatorKey words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular triangle has n = 3 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular triangle has 3 equal exterior angles.360 ÷ 3 = 120°
Check. The interior angle is 180 − 120 = 60°, and (3 − 2) × 180 ÷ 3 = 60° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular quadrilateral has n = 4 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular quadrilateral has 4 equal exterior angles.360 ÷ 4 = 90°
Check. The interior angle is 180 − 90 = 90°, and (4 − 2) × 180 ÷ 4 = 90° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular pentagon has n = 5 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular pentagon has 5 equal exterior angles.360 ÷ 5 = 72°
Check. The interior angle is 180 − 72 = 108°, and (5 − 2) × 180 ÷ 5 = 108° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular hexagon has n = 6 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular hexagon has 6 equal exterior angles.360 ÷ 6 = 60°
Check. The interior angle is 180 − 60 = 120°, and (6 − 2) × 180 ÷ 6 = 120° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular octagon has n = 8 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular octagon has 8 equal exterior angles.360 ÷ 8 = 45°
Check. The interior angle is 180 − 45 = 135°, and (8 − 2) × 180 ÷ 8 = 135° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular decagon has n = 10 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular decagon has 10 equal exterior angles.360 ÷ 10 = 36°
Check. The interior angle is 180 − 36 = 144°, and (10 − 2) × 180 ÷ 10 = 144° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 60°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 60°, so the number of them is 360 ÷ 60.360 ÷ 60 = 6 sides
Check. 6 × 60 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 45°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 45°, so the number of them is 360 ÷ 45.360 ÷ 45 = 8 sides
Check. 8 × 45 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 40°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 40°, so the number of them is 360 ÷ 40.360 ÷ 40 = 9 sides
Check. 9 × 40 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 30°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 30°, so the number of them is 360 ÷ 30.360 ÷ 30 = 12 sides
Check. 12 × 30 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 20°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 20°, so the number of them is 360 ÷ 20.360 ÷ 20 = 18 sides
Check. 18 × 20 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. Divide 360° by one exterior angle to find how many there are.
- Write down what we know. The polygon is regular, so all its exterior angles are equal.each exterior angle = 15°
- Recall the fact. The number of exterior angles equals the number of sides.The exterior angles of any polygon add up to 360°.
- Divide. In a regular polygon every exterior angle is 15°, so the number of them is 360 ÷ 15.360 ÷ 15 = 24 sides
Check. 24 × 15 = 360°, which is 360°. ✓
Watch out. If the question gives the interior angle, find the exterior angle first (180° minus the interior angle).
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 120°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 120 = 60°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 60° angles are equal, so divide 360° by the exterior angle.360 ÷ 60 = 6 sides
Check. Each interior angle of a regular hexagon is (6 − 2) × 180 ÷ 6 = 120°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 144°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 144 = 36°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 36° angles are equal, so divide 360° by the exterior angle.360 ÷ 36 = 10 sides
Check. Each interior angle of a regular decagon is (10 − 2) × 180 ÷ 10 = 144°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 150°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 150 = 30°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 30° angles are equal, so divide 360° by the exterior angle.360 ÷ 30 = 12 sides
Check. Each interior angle of a regular dodecagon is (12 − 2) × 180 ÷ 12 = 150°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 160°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 160 = 20°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 20° angles are equal, so divide 360° by the exterior angle.360 ÷ 20 = 18 sides
Check. Each interior angle of a regular 18-sided polygon is (18 − 2) × 180 ÷ 18 = 160°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior and exterior angles at a corner lie on a straight line. Find the exterior angle, then divide 360° by it.
- Write down what we know. The polygon is regular, so all its interior angles are equal, and so are all its exterior angles.each interior angle = 168°
- Find the exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.180 − 168 = 12°
- Use the sum of the exterior angles. The exterior angles of any polygon add up to 360°. In a regular polygon all 12° angles are equal, so divide 360° by the exterior angle.360 ÷ 12 = 30 sides
Check. Each interior angle of a regular 30-sided polygon is (30 − 2) × 180 ÷ 30 = 168°. ✓
Watch out. Do not divide the interior angle into 360 directly: use the exterior angle.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular 15-sided polygon has n = 15 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(15 − 2) × 180 = 13 × 180 = 2340°
- Share the sum equally. In a regular polygon all 15 interior angles are equal, so each one is the sum divided by 15.2340 ÷ 15 = 156°
Check. Another way: each exterior angle is 360 ÷ 15 = 24°, and 180 − 24 = 156°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal.
- Note what the word regular tells us. A regular polygon has equal sides and equal angles, so all the exterior angles are the same size.a regular octagon has n = 8 sides
- Recall the fact. If you walk round a polygon and turn at each corner by the exterior angle, you make one full turn, 360°, by the time you are facing the way you started.The exterior angles of any polygon add up to 360°.
- Share equally. A regular octagon has 8 equal exterior angles.360 ÷ 8 = 45°
Check. The interior angle is 180 − 45 = 135°, and (8 − 2) × 180 ÷ 8 = 135° as well. ✓
Watch out. Do not divide 180° by n. The exterior angles add up to 360°.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the sum of the interior angles, then divide by the number of angles because they are all equal.
- Note what the word regular tells us. A regular polygon has all its sides equal and all its angles equal, so the angles can be shared out equally.a regular octagon has n = 8 sides
- Find the sum of the interior angles. An n-sided polygon can be split into n − 2 triangles by joining one corner to all the others. Each triangle has angles adding up to 180°.(8 − 2) × 180 = 6 × 180 = 1080°
- Share the sum equally. In a regular polygon all 8 interior angles are equal, so each one is the sum divided by 8.1080 ÷ 8 = 135°
Check. Another way: each exterior angle is 360 ÷ 8 = 45°, and 180 − 45 = 135°. ✓
Watch out. This only works for a regular polygon, where all the angles are the same.
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The number of sides is 360 divided by the exterior angle, and it must be a whole number.
- Recall the fact. The exterior angles of any polygon add up to 360°.number of sides = 360 ÷ exterior angle
- Divide. 360 ÷ 50 = 7.2
- Decide. A polygon cannot have 7.2 sides.No
Check. Exterior angles that work are factors of 360: for example 45° (8 sides) or 40° (9 sides). ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Find the interior angle of each shape, add them, and subtract from 360° (angles around a point).
- Interior angle of the pentagon. Use (n − 2) × 180 ÷ n.3 × 180 ÷ 5 = 108°
- Interior angle of the hexagon. 4 × 180 ÷ 6 = 120°
- Add them. 108 + 120 = 228°
- Subtract from the full turn. Angles around a point add up to 360°.360 − 228 = 132°
Check. 132 + 108 + 120 = 360°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The three interior angles must add up to 360° (angles around a point).
- Square and hexagon. Interior angles: 90° and 120°.90 + 120 = 210°
- Find the angle left. Angles around a point add up to 360°.360 − 210 = 150°
- Find n from the interior angle. The exterior angle is 180 − 150 = 30°, so n = 360 ÷ 30.n = 12
Check. A regular 12-sided polygon has interior angle 10 × 180 ÷ 12 = 150°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. The interior angles of the tiles at a point must add up to exactly 360°.
- Hexagons. The interior angle is 120°.3 × 120 = 360°: they fit exactly
- Pentagons. The interior angle is 108°.3 × 108 = 324°: gap of 36°
4 × 108 = 432°: too much (overlap) - Decide. No whole number of pentagon corners makes exactly 360°.Pentagons cannot tile a floor without gaps.
Check. 360 ÷ 108 = 3.33..., not a whole number. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Call the exterior angle e. Then the interior angle is 5e, and they add up to 180°.
- Write the interior angle in terms of e. The exterior is one fifth of the interior, so the interior is five times the exterior.interior = 5e
- Use the straight line. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.5e + e = 180, so 6e = 180
- Solve. Divide both sides by 6.e = 30°
- Find the number of sides. The exterior angles of any polygon add up to 360°.360 ÷ 30 = 12
Check. Interior angle = 150° and 150 ÷ 5 = 30°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Interior + exterior = 180°. Form an equation, solve it, then use 360 ÷ exterior angle.
- Write the equation. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.(4x + 10) + (x − 5) = 180
- Collect like terms. 5x + 5 = 180
- Subtract 5. 5x = 175
- Divide by 5. x = 35
- Find the exterior angle. Substitute x = 35.35 − 5 = 30°
- Find the number of sides. The exterior angles of any polygon add up to 360°.360 ÷ 30 = 12
Check. Interior = 4 × 35 + 10 = 150°, and 150 + 30 = 180°. ✓
Key words. A polygon is a flat shape with straight sides. An interior angle is inside the shape at a corner. An exterior angle is formed outside the shape when one side is extended. In a regular polygon all the sides are equal and all the angles are equal.
Method. Write each exterior angle as 180° minus its interior angle, then add them.
- Write each exterior angle. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.exterior angles: 180 − a, 180 − b, 180 − c
- Add them. (180 − a) + (180 − b) + (180 − c) = 540 − (a + b + c)
- Use the angle sum of the triangle. Angles in a triangle add up to 180°.a + b + c = 180
- Finish. 540 − 180 = 360° ■
Check. For an equilateral triangle: 3 × 120 = 360°. ✓