GCSE Maths: The angle in a semicircle (Higher)
Part of the GCSE Maths circle theorems chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
The angle in a semicircle
Draw any diameter AB of a circle, then pick any point C on the circle and join it to A and to B. Whichever point C you choose, the angle at C is always a right angle.
This is the angle in a semicircle theorem. The name comes from the diameter AB cutting the circle into two semicircles, and C being a point on the curved edge of one of them. Reason to write: “angle in a semicircle is 90°”. It is one of the most useful facts: a right angle lets you use Pythagoras' theorem and trigonometry (Chapter 20).
Why is it true? Join C to the centre O. Now OA, OB and OC are all radii, so they are equal. That makes two isosceles triangles, OAC and OBC. Call the equal base angles a (in triangle OAC) and b (in triangle OBC). The angle at C in the big triangle is a + b. The angles of triangle ABC add up to 180°, so a + b + (a + b) = 180°, which means 2(a + b) = 180° and a + b = 90°. You will see this written out as a proof in worked example 26.2.4.
Key ideas
- If AB is a diameter and C is on the circle, then angle ACB = 90°.
- Reason: “angle in a semicircle is 90°”.
- The right angle is at the point on the circle, opposite the diameter.
- The right-angled triangle ABC has the diameter as its hypotenuse (longest side).
- The converse is also true: if a triangle with a right angle at C is drawn in a circle through A, B and C, then AB is a diameter.
Watch out
The theorem only works when the line is a diameter. If the line AB does not pass through the centre, the angle at C is not 90°. Always check for the centre O (or the word “diameter”) before you use it.Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Mark the triangle ABC. The diameter AB and the point C on the circle make a triangle. We are looking for angle ABC in it.Angle CAB = 35°, angle ABC = ?
- Find angle ACB. The line AB is a diameter, so the angle at C on the circle is an angle in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Use the angle sum of the triangle. The three angles of triangle ABC add up to 180°, so subtract the two we know.Angle ABC = 180° − 90° − 35°
Angle ABC = 55° (reason: the angles in a triangle add up to 180°)
Check. 35° + 55° + 90° = 180°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Find angle ACB. The line AB is a diameter, so the angle at C is in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Find angle OCB. OC and OB are both radii, so triangle OCB is isosceles. Its base angles are equal.Angle OCB = angle OBC = 66° (reason: the base angles of an isosceles triangle are equal)
- Subtract to get angle ACO. Angle ACB is made of angle ACO and angle OCB side by side.Angle ACO = 90° − 66° = 24°
Check. Triangle OAC is also isosceles (OA = OC) with angle OAC = 24°, so angle ACO = 24°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Method. The angle in a semicircle is 90°, so the other two angles of the triangle add up to 90°. That gives an equation in x.
- Find angle ACB. The line AB is a diameter, so angle ACB is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write an equation. The three angles of triangle ABC add up to 180°. With the right angle taken away, the other two must add to 90°.(3x + 10) + (3x + 20) = 90
- Collect the like terms. Add the x terms together and the numbers together.6x + 30 = 90
- Subtract 30 from both sides. We want the number terms on the other side.6x = 90 − 30
6x = 60 - Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 60 ÷ 6 = 10
- Check the answer by working out both angles. Put x back into each angle. They should add up to 90°.3 × 10 + 10 = 40° and 3 × 10 + 20 = 50°
40° + 50° = 90° ✓
Check. The right angle plus 40° plus 50° is 90° + 90° = 180°. ✓
Watch out. The equation is "angles add to 90°", not 180°, because the right angle is already used up.
Key words. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Method. A diameter gives a right angle at the circle, so Pythagoras' theorem can be used in triangle ABC.
- Find the right angle. AB is a diameter, so the angle at C is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write Pythagoras for triangle ABC. The longest side (the hypotenuse) is opposite the right angle. It is AB, the diameter.AB2 = AC2 + BC2
172 = 82 + BC2 - Work out the squares. 289 = 64 + BC2
- Rearrange for BC2. Subtract the known square from both sides.BC2 = 289 − 64 = 225
- Square root. Lengths are positive, so take the positive root.BC = √225 = 15 cm
Check. 82 + 152 = 64 + 225 = 289 = 172. ✓
Watch out. Pythagoras: the diameter is the longest side (hypotenuse). Never add the squares of the hypotenuse.
Method. Join C to the centre O to make two isosceles triangles, then use the angle sum of triangle ABC.
- Draw the radius OC. This splits the triangle into two smaller triangles that we can reason about.OA = OC = OB (all radii of the circle)
- Label the base angles in triangle OAC. It is isosceles because OA = OC, so its base angles are equal. Call each one a.angle OAC = angle OCA = a
- Label the base angles in triangle OBC. It is isosceles because OB = OC. Call the base angles b.angle OBC = angle OCB = b
- Write angle ACB. It is made of the two angles at C side by side.angle ACB = a + b
- Use the angle sum in triangle ABC. The angles of a triangle add up to 180°. The three angles are a at A, b at B and a + b at C.a + b + (a + b) = 180°
2(a + b) = 180° - Divide by 2 and conclude. Divide both sides by 2.a + b = 90°, so angle ACB = 90°. The angle in a semicircle is a right angle. ✓
Check. Try numbers: if a = 25° and b = 65° then angle ACB = 90°, and the triangle angles are 25°, 65° and 90°, which add up to 180°. ✓
Watch out. A proof must work for every position of C, so use letters (a and b) instead of numbers.
Exercise 26B The angle in a semicircle
Non-calculatorKey words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Mark the triangle ABC. The diameter AB and the point C on the circle make a triangle. We are looking for angle ABC in it.Angle CAB = 25°, angle ABC = ?
- Find angle ACB. The line AB is a diameter, so the angle at C on the circle is an angle in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Use the angle sum of the triangle. The three angles of triangle ABC add up to 180°, so subtract the two we know.Angle ABC = 180° − 90° − 25°
Angle ABC = 65° (reason: the angles in a triangle add up to 180°)
Check. 25° + 65° + 90° = 180°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Mark the triangle ABC. The diameter AB and the point C on the circle make a triangle. We are looking for angle ABC in it.Angle CAB = 38°, angle ABC = ?
- Find angle ACB. The line AB is a diameter, so the angle at C on the circle is an angle in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Use the angle sum of the triangle. The three angles of triangle ABC add up to 180°, so subtract the two we know.Angle ABC = 180° − 90° − 38°
Angle ABC = 52° (reason: the angles in a triangle add up to 180°)
Check. 38° + 52° + 90° = 180°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Mark the triangle ABC. The diameter AB and the point C on the circle make a triangle. We are looking for angle ABC in it.Angle CAB = 57°, angle ABC = ?
- Find angle ACB. The line AB is a diameter, so the angle at C on the circle is an angle in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Use the angle sum of the triangle. The three angles of triangle ABC add up to 180°, so subtract the two we know.Angle ABC = 180° − 90° − 57°
Angle ABC = 33° (reason: the angles in a triangle add up to 180°)
Check. 57° + 33° + 90° = 180°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Mark the triangle ABC. The diameter AB and the point C on the circle make a triangle. We are looking for angle ABC in it.Angle CAB = 64°, angle ABC = ?
- Find angle ACB. The line AB is a diameter, so the angle at C on the circle is an angle in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Use the angle sum of the triangle. The three angles of triangle ABC add up to 180°, so subtract the two we know.Angle ABC = 180° − 90° − 64°
Angle ABC = 26° (reason: the angles in a triangle add up to 180°)
Check. 64° + 26° + 90° = 180°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Find angle ACB. The line AB is a diameter, so the angle at C is in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Find angle OCB. OC and OB are both radii, so triangle OCB is isosceles. Its base angles are equal.Angle OCB = angle OBC = 70° (reason: the base angles of an isosceles triangle are equal)
- Subtract to get angle ACO. Angle ACB is made of angle ACO and angle OCB side by side.Angle ACO = 90° − 70° = 20°
Check. Triangle OAC is also isosceles (OA = OC) with angle OAC = 20°, so angle ACO = 20°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Find angle ACB. The line AB is a diameter, so the angle at C is in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Find angle OCB. OC and OB are both radii, so triangle OCB is isosceles. Its base angles are equal.Angle OCB = angle OBC = 57° (reason: the base angles of an isosceles triangle are equal)
- Subtract to get angle ACO. Angle ACB is made of angle ACO and angle OCB side by side.Angle ACO = 90° − 57° = 33°
Check. Triangle OAC is also isosceles (OA = OC) with angle OAC = 33°, so angle ACO = 33°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Key words. A diameter is a chord through the centre. An angle in a semicircle is the angle made at the circle by lines drawn to both ends of a diameter.
Method. A diameter makes a right angle with every point on the circle. Use that right angle, then the angle sum of a triangle.
- Find angle ACB. The line AB is a diameter, so the angle at C is in a semicircle.Angle ACB = 90° (reason: angle in a semicircle is 90°)
- Find angle OCB. OC and OB are both radii, so triangle OCB is isosceles. Its base angles are equal.Angle OCB = angle OBC = 42° (reason: the base angles of an isosceles triangle are equal)
- Subtract to get angle ACO. Angle ACB is made of angle ACO and angle OCB side by side.Angle ACO = 90° − 42° = 48°
Check. Triangle OAC is also isosceles (OA = OC) with angle OAC = 48°, so angle ACO = 48°. ✓
Watch out. The right angle is at the point on the circle (C), not at the end of the diameter.
Method. The angle in a semicircle is 90°, so the other two angles of the triangle add up to 90°. That gives an equation in x.
- Find angle ACB. The line AB is a diameter, so angle ACB is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write an equation. The three angles of triangle ABC add up to 180°. With the right angle taken away, the other two must add to 90°.(3x + 10) + (3x + 20) = 90
- Collect the like terms. Add the x terms together and the numbers together.6x + 30 = 90
- Subtract 30 from both sides. We want the number terms on the other side.6x = 90 − 30
6x = 60 - Divide both sides by 6. 6x means 6 × x, so dividing by 6 leaves x on its own.x = 60 ÷ 6 = 10
- Check the answer by working out both angles. Put x back into each angle. They should add up to 90°.3 × 10 + 10 = 40° and 3 × 10 + 20 = 50°
40° + 50° = 90° ✓
Check. The right angle plus 40° plus 50° is 90° + 90° = 180°. ✓
Watch out. The equation is "angles add to 90°", not 180°, because the right angle is already used up.
Method. The angle in a semicircle is 90°, so the other two angles of the triangle add up to 90°. That gives an equation in x.
- Find angle ACB. The line AB is a diameter, so angle ACB is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write an equation. The three angles of triangle ABC add up to 180°. With the right angle taken away, the other two must add to 90°.(5x + 10) + (4x + 26) = 90
- Collect the like terms. Add the x terms together and the numbers together.9x + 36 = 90
- Subtract 36 from both sides. We want the number terms on the other side.9x = 90 − 36
9x = 54 - Divide both sides by 9. 9x means 9 × x, so dividing by 9 leaves x on its own.x = 54 ÷ 9 = 6
- Check the answer by working out both angles. Put x back into each angle. They should add up to 90°.5 × 6 + 10 = 40° and 4 × 6 + 26 = 50°
40° + 50° = 90° ✓
Check. The right angle plus 40° plus 50° is 90° + 90° = 180°. ✓
Watch out. The equation is "angles add to 90°", not 180°, because the right angle is already used up.
Method. The angle in a semicircle is 90°, so the other two angles of the triangle add up to 90°. That gives an equation in x.
- Find angle ACB. The line AB is a diameter, so angle ACB is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write an equation. The three angles of triangle ABC add up to 180°. With the right angle taken away, the other two must add to 90°.(2x + 14) + (5x + 20) = 90
- Collect the like terms. Add the x terms together and the numbers together.7x + 34 = 90
- Subtract 34 from both sides. We want the number terms on the other side.7x = 90 − 34
7x = 56 - Divide both sides by 7. 7x means 7 × x, so dividing by 7 leaves x on its own.x = 56 ÷ 7 = 8
- Check the answer by working out both angles. Put x back into each angle. They should add up to 90°.2 × 8 + 14 = 30° and 5 × 8 + 20 = 60°
30° + 60° = 90° ✓
Check. The right angle plus 30° plus 60° is 90° + 90° = 180°. ✓
Watch out. The equation is "angles add to 90°", not 180°, because the right angle is already used up.
Key words. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Method. A diameter gives a right angle at the circle, so Pythagoras' theorem can be used in triangle ABC.
- Find the right angle. AB is a diameter, so the angle at C is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write Pythagoras for triangle ABC. The longest side (the hypotenuse) is opposite the right angle. It is AB, the diameter.AB2 = AC2 + BC2
102 = 62 + BC2 - Work out the squares. 100 = 36 + BC2
- Rearrange for BC2. Subtract the known square from both sides.BC2 = 100 − 36 = 64
- Square root. Lengths are positive, so take the positive root.BC = √64 = 8 cm
Check. 62 + 82 = 36 + 64 = 100 = 102. ✓
Watch out. Pythagoras: the diameter is the longest side (hypotenuse). Never add the squares of the hypotenuse.
Key words. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Method. A diameter gives a right angle at the circle, so Pythagoras' theorem can be used in triangle ABC.
- Find the right angle. AB is a diameter, so the angle at C is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write Pythagoras for triangle ABC. The longest side (the hypotenuse) is opposite the right angle. It is AB, the diameter.AB2 = AC2 + BC2
172 = 152 + BC2 - Work out the squares. 289 = 225 + BC2
- Rearrange for BC2. Subtract the known square from both sides.BC2 = 289 − 225 = 64
- Square root. Lengths are positive, so take the positive root.BC = √64 = 8 cm
Check. 152 + 82 = 225 + 64 = 289 = 172. ✓
Watch out. Pythagoras: the diameter is the longest side (hypotenuse). Never add the squares of the hypotenuse.
Key words. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Method. A diameter gives a right angle at the circle, so Pythagoras' theorem can be used in triangle ABC.
- Find the right angle. AB is a diameter, so the angle at C is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Write Pythagoras for triangle ABC. The longest side (the hypotenuse) is opposite the right angle. It is AB, the diameter.AB2 = AC2 + BC2
292 = 202 + BC2 - Work out the squares. 841 = 400 + BC2
- Rearrange for BC2. Subtract the known square from both sides.BC2 = 841 − 400 = 441
- Square root. Lengths are positive, so take the positive root.BC = √441 = 21 cm
Check. 202 + 212 = 400 + 441 = 841 = 292. ✓
Watch out. Pythagoras: the diameter is the longest side (hypotenuse). Never add the squares of the hypotenuse.
Key words. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Method. A diameter gives a right angle at the circle, so Pythagoras' theorem can be used in triangle ABC.
- Find the right angle. AB is a diameter, so angle ACB is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Use Pythagoras to find the diameter AB. Triangle ABC is right-angled and AB is its longest side.AB2 = 92 + 122 = 81 + 144 = 225
- Square root. The diameter is the positive root.AB = √225 = 15 cm
- Halve the diameter. The radius is half of the diameter.radius = 15 ÷ 2 = 7.5 cm
Check. Diameter = 2 × 7.5 = 15 cm, and 92 + 122 = 225 = 152. ✓
Watch out. Pythagoras: the diameter is the longest side (hypotenuse). Never add the squares of the hypotenuse.
Key words. The hypotenuse is the longest side of a right-angled triangle, opposite the right angle.
Method. A diameter gives a right angle at the circle, so Pythagoras' theorem can be used in triangle ABC.
- Find the right angle. AB is a diameter, so angle ACB is an angle in a semicircle.Angle ACB = 90° (angle in a semicircle is 90°)
- Use Pythagoras to find the diameter AB. Triangle ABC is right-angled and AB is its longest side.AB2 = 72 + 242 = 49 + 576 = 625
- Square root. The diameter is the positive root.AB = √625 = 25 cm
- Halve the diameter. The radius is half of the diameter.radius = 25 ÷ 2 = 12.5 cm
Check. Diameter = 2 × 12.5 = 25 cm, and 72 + 242 = 625 = 252. ✓
Watch out. Pythagoras: the diameter is the longest side (hypotenuse). Never add the squares of the hypotenuse.
Method. Recall the exact conditions of the theorem.
- Recall the theorem. The angle in a semicircle is 90° only when the two arms of the angle go to the two ends of a diameter.angle ACB = 90° if AB is a diameter
- Compare with Mia's reason. Mia only said C is on the circle. If AB is not a diameter, the angle at C can be any size.Missing: AB is a diameter (it passes through the centre).
Check. In a circle, if AB is not a diameter then angle ACB changes as C moves round the circle. ✓
Watch out. Always say what makes the right angle: the line must be a diameter.
Method. The converse of the semicircle theorem: if a triangle in a circle has a right angle, the hypotenuse is a diameter.
- Find the hypotenuse. The longest side of a right-angled triangle is opposite the right angle.hypotenuse = 15 cm (check: 92 + 122 = 81 + 144 = 225 = 152)
- Use the circle theorem. A right angle on the circle means the side opposite it passes through the centre, so it is a diameter.diameter = 15 cm
- Halve the diameter. The radius is half of the diameter.radius = 15 ÷ 2 = 7.5 cm
Check. A circle of radius 7.5 cm has a diameter of 15 cm, the same as the hypotenuse. ✓
Watch out. The right angle tells you which side is the diameter. It is the side opposite the right angle.
Method. Use the angle at the centre being twice the angle at the circumference.
- Name the arc. Angle ACB is at the circumference and stands on the arc AB (the arc not containing C).angle at the circumference = 90°
- Double it. The angle at the centre is twice the angle at the circumference.angle AOB = 2 × 90° = 180°
- Conclude. An angle of 180° is a straight line, so A, O and B are in a straight line and AB passes through the centre.So AB is a diameter. ✓
Check. This matches the known fact: a right-angled triangle drawn in a circle has its hypotenuse as a diameter. ✓
Watch out. This proof uses a theorem from the next section. You can also prove it by drawing a rectangle.
Method. Use the isosceles triangles to name every angle, then the angle sum.
- Angle at A. It is a base angle of triangle OAC.angle CAB = a
- Angle at B. It is a base angle of triangle OBC.angle CBA = b
- Angle at C. It is made of the two angles at C.angle ACB = a + b
- Angle sum. The angles of a triangle add up to 180°.a + b + a + b = 180°, so 2(a + b) = 180° and a + b = 90°. So angle ACB = 90°.
Check. Test with a = 30°, b = 60°. Then 30 + 60 + 90 = 180. ✓
Watch out. Write each fact and its reason as a separate line.