GCSE Maths: Area of a triangle: ½ab sin C (Higher)
Part of the GCSE Maths pythagoras and trigonometry chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Area of a triangle: ½ab sin C
You already know that the area of a triangle is ½ × base × height. But sometimes the height is not given. If you know two sides and the angle between them, you can still find the area.
Area = ½ ab sin C
Here a and b are two sides and C is the angle between them.
Why it works. Take a as the base. Draw the height h from the opposite vertex. In the right-angled triangle, the hypotenuse is b and the angle at C is C, so sin C = h ÷ b, which gives h = b sin C. Then area = ½ × base × height = ½ × a × b sin C. That is the formula.
The formula can be used to find the area, but it can also be rearranged to find a side or an angle when the area is known. To find an angle, you get sin C. As for the sine rule, there are two angles between 0° and 180° with the same sine, an acute one and an obtuse one (180° minus the acute one). Read the question to see which is wanted, or give both.
The formula is also useful for polygons. A regular polygon with n sides splits into n identical isosceles triangles meeting at the centre. Each has two equal sides (the radius) and an angle of 360° ÷ n between them, so its area is ½ r2 sin(360° ÷ n).
Key ideas
- Area = ½ ab sin C, where C is the angle between sides a and b.
- To find an angle: sin C = 2 × area ÷ (ab). Remember the possible obtuse angle 180° − C.
- To find a side, rearrange: a = 2 × area ÷ (b sin C).
- Regular polygons: n triangles, each with area ½ r2 sin(360° ÷ n).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Area of any triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
- Spot the two sides and the angle between them. The formula needs two sides and the included angle (the angle between those two sides).Sides: 8 cm and 11 cm. Included angle C = 40°.
- Write the formula and put the numbers in. Area = half × one side × the other side × sine of the angle between them.Area = ½ ab sin C
Area = ½ × 8 × 11 × sin 40° - Use the calculator. Find the sine first, then multiply everything together.sin 40° = 0.64278…
Area = 0.5 × 8 × 11 × 0.64278… = 28.28265… - Round to 1 d.p. and write the units. Area is measured in square units.Area = 28.3 cm2
Check. Another way: take 8 as the base. The height is 11 × sin 40° = 7.07066…. Area = ½ × 8 × 7.07066… = 28.28265…. ✓
Watch out. The angle must be the one between the two sides you use. If it is not between them, find the angle you need first (angles in a triangle add up to 180°).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C, rearrange to find sin C, then use the inverse sine.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here the angle C is unknown.30 = ½ × 8 × 11 × sin C
- Work out ½ × a × b. This is a plain number, so we can combine it.½ × 8 × 11 = 44
30 = 44 sin C - Divide both sides by 44. This gets sin C on its own.sin C = 3044 = 0.68181…
- Use the inverse sine. The inverse button turns the sine back into an angle.C = sin−1(0.68181…) = 42.98588…°
- Think about a second (obtuse) angle. Sine of an obtuse angle equals sine of 180° minus it, so both give the same area.Second angle = 180° − 42.98588…° = 137.01411…°
- Round to 1 d.p. Both angles give the same area, so both are answers.C = 43.0° or 137.0°
Check. Work forwards: ½ × 8 × 11 × sin 43.0° = 30.0 cm2, close to 30. ✓
Watch out. The inverse sine only gives the acute angle. An obtuse angle gives the same sine, so check whether the question wants it.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C and rearrange to find the missing side.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here one side is unknown.35 = ½ × a × 10 × sin 48°
- Work out the known numbers. Combine ½, the side and sin C into one number.sin 48° = 0.74314…
½ × 10 × 0.74314… = 3.71572…
35 = 3.71572… a - Divide both sides by 3.71572…. This leaves a on its own.a = 35 ÷ 3.71572… = 9.41942…
- Round to 1 d.p. Give the unit.a = 9.4 cm
Check. Work forwards: ½ × 9.4 × 10 × sin 48° = 34.9 cm2, close to 35. ✓
Watch out. Be careful rearranging: divide by everything that multiplies the unknown side (the ½, the other side and sin C).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. A regular octagon is made of 8 identical triangles. Find the area of one triangle with ½ ab sin C and multiply by 8.
- Find the angle at the centre. The 8 angles at the centre make a full turn of 360°.360° ÷ 8 = 45°
- Find the area of one triangle. Two sides are the radius 10 cm, and the angle between them is 45°.½ × 10 × 10 × sin 45° = 50 × 0.70710… = 35.35533… cm2
- Multiply by 8. There are 8 identical triangles.8 × 35.35533… = 282.84271…
- Round. Area = 282.8 cm2
Check. Exact version: 8 × 50 × √2÷2 = 200√2 = 282.84271…. ✓
Watch out. Multiply by the number of triangles at the end.
Watch out
The angle in the formula must be the one between the two sides you use. If it is not, you must find the angle you need first, using the fact that angles in a triangle add to 180°. Also remember the units: area is in square units, such as cm2.Exercise 20L Area of a triangle using ½ab sin C
Calculator allowedKey words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Area of any triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
- Spot the two sides and the angle between them. The formula needs two sides and the included angle (the angle between those two sides).Sides: 6 cm and 9 cm. Included angle C = 35°.
- Write the formula and put the numbers in. Area = half × one side × the other side × sine of the angle between them.Area = ½ ab sin C
Area = ½ × 6 × 9 × sin 35° - Use the calculator. Find the sine first, then multiply everything together.sin 35° = 0.57357…
Area = 0.5 × 6 × 9 × 0.57357… = 15.48656… - Round to 1 d.p. and write the units. Area is measured in square units.Area = 15.5 cm2
Check. Another way: take 6 as the base. The height is 9 × sin 35° = 5.16218…. Area = ½ × 6 × 5.16218… = 15.48656…. ✓
Watch out. The angle must be the one between the two sides you use. If it is not between them, find the angle you need first (angles in a triangle add up to 180°).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Area of any triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
- Spot the two sides and the angle between them. The formula needs two sides and the included angle (the angle between those two sides).Sides: 10.4 cm and 7.5 cm. Included angle C = 62°.
- Write the formula and put the numbers in. Area = half × one side × the other side × sine of the angle between them.Area = ½ ab sin C
Area = ½ × 10.4 × 7.5 × sin 62° - Use the calculator. Find the sine first, then multiply everything together.sin 62° = 0.88294…
Area = 0.5 × 10.4 × 7.5 × 0.88294… = 34.43495… - Round to 1 d.p. and write the units. Area is measured in square units.Area = 34.4 cm2
Check. Another way: take 10.4 as the base. The height is 7.5 × sin 62° = 6.62210…. Area = ½ × 10.4 × 6.62210… = 34.43495…. ✓
Watch out. The angle must be the one between the two sides you use. If it is not between them, find the angle you need first (angles in a triangle add up to 180°).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Area of any triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
- Spot the two sides and the angle between them. The formula needs two sides and the included angle (the angle between those two sides).Sides: 5.2 cm and 8.8 cm. Included angle C = 118°.
- Write the formula and put the numbers in. Area = half × one side × the other side × sine of the angle between them.Area = ½ ab sin C
Area = ½ × 5.2 × 8.8 × sin 118° - Use the calculator. Find the sine first, then multiply everything together.sin 118° = 0.88294…
Area = 0.5 × 5.2 × 8.8 × 0.88294… = 20.20184… - Round to 1 d.p. and write the units. Area is measured in square units.Area = 20.2 cm2
Check. Another way: take 5.2 as the base. The height is 8.8 × sin 118° = 7.76993…. Area = ½ × 5.2 × 7.76993… = 20.20184…. ✓
Watch out. The angle must be the one between the two sides you use. If it is not between them, find the angle you need first (angles in a triangle add up to 180°).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Area of any triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
- Spot the two sides and the angle between them. The formula needs two sides and the included angle (the angle between those two sides).Sides: 12 cm and 12 cm. Included angle C = 25°.
- Write the formula and put the numbers in. Area = half × one side × the other side × sine of the angle between them.Area = ½ ab sin C
Area = ½ × 12 × 12 × sin 25° - Use the calculator. Find the sine first, then multiply everything together.sin 25° = 0.42261…
Area = 0.5 × 12 × 12 × 0.42261… = 30.42851… - Round to 1 d.p. and write the units. Area is measured in square units.Area = 30.4 cm2
Check. Another way: take 12 as the base. The height is 12 × sin 25° = 5.07141…. Area = ½ × 12 × 5.07141… = 30.42851…. ✓
Watch out. The angle must be the one between the two sides you use. If it is not between them, find the angle you need first (angles in a triangle add up to 180°).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C, rearrange to find sin C, then use the inverse sine.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here the angle C is unknown.20 = ½ × 8 × 7 × sin C
- Work out ½ × a × b. This is a plain number, so we can combine it.½ × 8 × 7 = 28
20 = 28 sin C - Divide both sides by 28. This gets sin C on its own.sin C = 2028 = 0.71428…
- Use the inverse sine. The inverse button turns the sine back into an angle.C = sin−1(0.71428…) = 45.58469…°
- Think about a second (obtuse) angle. Sine of an obtuse angle equals sine of 180° minus it, so both give the same area.Second angle = 180° − 45.58469…° = 134.41530…°
- Choose the acute angle, round to 1 d.p. The question tells us the angle is acute.C = 45.6°
Check. Work forwards: ½ × 8 × 7 × sin 45.6° = 20.0 cm2, close to 20. ✓
Watch out. The inverse sine only gives the acute angle. An obtuse angle gives the same sine, so check whether the question wants it.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C, rearrange to find sin C, then use the inverse sine.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here the angle C is unknown.30 = ½ × 8 × 11 × sin C
- Work out ½ × a × b. This is a plain number, so we can combine it.½ × 8 × 11 = 44
30 = 44 sin C - Divide both sides by 44. This gets sin C on its own.sin C = 3044 = 0.68181…
- Use the inverse sine. The inverse button turns the sine back into an angle.C = sin−1(0.68181…) = 42.98588…°
- Think about a second (obtuse) angle. Sine of an obtuse angle equals sine of 180° minus it, so both give the same area.Second angle = 180° − 42.98588…° = 137.01411…°
- Choose the acute angle, round to 1 d.p. The question tells us the angle is acute.C = 43.0°
Check. Work forwards: ½ × 8 × 11 × sin 43.0° = 30.0 cm2, close to 30. ✓
Watch out. The inverse sine only gives the acute angle. An obtuse angle gives the same sine, so check whether the question wants it.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C and rearrange to find the missing side.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here one side is unknown.35 = ½ × a × 10 × sin 48°
- Work out the known numbers. Combine ½, the side and sin C into one number.sin 48° = 0.74314…
½ × 10 × 0.74314… = 3.71572…
35 = 3.71572… a - Divide both sides by 3.71572…. This leaves a on its own.a = 35 ÷ 3.71572… = 9.41942…
- Round to 1 d.p. Give the unit.a = 9.4 cm
Check. Work forwards: ½ × 9.4 × 10 × sin 48° = 34.9 cm2, close to 35. ✓
Watch out. Be careful rearranging: divide by everything that multiplies the unknown side (the ½, the other side and sin C).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C and rearrange to find the missing side.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here one side is unknown.22.5 = ½ × a × 7.5 × sin 90°
- Work out the known numbers. Combine ½, the side and sin C into one number.sin 90° = 1
½ × 7.5 × 1 = 3.75
22.5 = 3.75 a - Divide both sides by 3.75. This leaves a on its own.a = 22.5 ÷ 3.75 = 6
- Round to 1 d.p. Give the unit.a = 6.0 cm
Check. Work forwards: ½ × 6.0 × 7.5 × sin 90° = 22.5 cm2, close to 22.5. ✓
Watch out. Be careful rearranging: divide by everything that multiplies the unknown side (the ½, the other side and sin C).
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. A regular polygon with 8 sides is made of 8 identical triangles meeting at the centre.
- Find the angle at the centre. A full turn is 360°.360° ÷ 8 = 45°
- Find the area of one triangle. Two sides are the radius, with the angle between them: use ½ ab sin C.½ × 10 × 10 × sin 45° = 35.35533… cm2
- Multiply by 8. There are 8 triangles.8 × 35.35533… = 282.84271… cm2
- Round. Area = 282.8 cm2
Check. The polygon is inside a circle of radius 10 cm, whose area is π × 102 = 314.2 cm2. The polygon area 282.8 is just less. ✓
Watch out. Multiply by the number of sides at the end.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. A regular polygon with 5 sides is made of 5 identical triangles meeting at the centre.
- Find the angle at the centre. A full turn is 360°.360° ÷ 5 = 72°
- Find the area of one triangle. Two sides are the radius, with the angle between them: use ½ ab sin C.½ × 6 × 6 × sin 72° = 17.11901… cm2
- Multiply by 5. There are 5 triangles.5 × 17.11901… = 85.59508… cm2
- Round. Area = 85.6 cm2
Check. The polygon is inside a circle of radius 6 cm, whose area is π × 62 = 113.1 cm2. The polygon area 85.6 is just less. ✓
Watch out. Multiply by the number of sides at the end.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. A regular polygon with 12 sides is made of 12 identical triangles meeting at the centre.
- Find the angle at the centre. A full turn is 360°.360° ÷ 12 = 30°
- Find the area of one triangle. Two sides are the radius, with the angle between them: use ½ ab sin C.½ × 4 × 4 × sin 30° = 4 cm2
- Multiply by 12. There are 12 triangles.12 × 4 = 48 cm2
- Round. Area = 48.0 cm2
Check. The polygon is inside a circle of radius 4 cm, whose area is π × 42 = 50.3 cm2. The polygon area 48.0 is just less. ✓
Watch out. Multiply by the number of sides at the end.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. A parallelogram is made of two identical triangles. Find the area of one triangle with ½ ab sin C, then double it.
- Draw a diagonal. The diagonal cuts the parallelogram into two identical triangles. Each has sides 9 cm and 12 cm with the 65° angle between them.Triangle: 9 cm, 12 cm, 65°
- Area of one triangle. Use ½ ab sin C.½ × 9 × 12 × sin 65° = 54 × 0.90630… = 48.94062…
- Double it. There are two identical triangles.2 × 48.94062… = 97.881241
- Round. Area = 97.9 cm2
Check. The area of a parallelogram is base × height = 12 × (9 sin 65°) = 12 × 8.15677… = 97.881241. ✓
Watch out. The parallelogram is two triangles, so the area is ab sin C, with no half.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Use ½ ab sin C to find the area, then multiply by the cost for each square metre.
- Find the area. The two sides are 120 m and 95 m, and the angle between them is 70°.Area = ½ × 120 × 95 × sin 70° = 5700 × 0.93969… = 5356.24793… m2
- Work out the cost. Cost = area × price per m2. Use the unrounded area and round the money at the end.5356.24793… × 0.35 = 1874.68677…
- Round to the nearest pound. Cost = £1875
Check. Rounded area: 5400 × 0.35 = 1890, close to 1875. ✓
Watch out. Keep the full area in the calculator when you multiply by the price.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. We know three sides but no angle, so first use the cosine rule to find an angle, then use area = ½ ab sin C.
- Find the angle between the 7 cm and 8 cm sides. The side opposite this angle C is 9 cm. Use the rearranged cosine rule.cos C = 72 + 82 − 922 × 7 × 8 = 32112 = 0.28571…
C = cos−1(0.28571…) = 73.39845…° - Use the area formula. Keep the unrounded angle in your calculator.Area = ½ × 7 × 8 × sin 73.39845…° = 28 × 0.95831… = 26.83281…
- Round. Area = 26.8 cm2
Check. Heron’s formula check: s = (7 + 8 + 9) ÷ 2 = 12, and √(12 × 5 × 4 × 3) = √720 = 26.83281…. ✓
Watch out. Do not round the angle before using it in the area formula.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. The formula needs the angle between the two sides. Check where Sam’s angle is.
- Look at the diagram. The 50° angle is at A. It is between AB and AC, but Sam used the sides AB = 7 and BC = 9.The angle between AB and BC is the angle at B, not 50°.
- Say what should be done. Either use the two sides AB and AC with the 50° angle, or find the angle B first.Sam used the wrong angle.
Check. The sides AB and BC meet at B. The angle between them is angle B. ✓
Watch out. Write the two sides and the angle between them before using the formula.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Use area = ½ ab sin C with the exact value sin 60° = √3 ÷ 2.
- Write the formula. All three sides are 6 cm and each angle is 60°.Area = ½ × 6 × 6 × sin 60°
- Put in the exact value. sin 60° = √3 ÷ 2.Area = ½ × 6 × 6 × √32 = 18 × √32
- Simplify. 18 ÷ 2 = 9.Area = 9√3 cm2
Check. Decimal check: 9√3 = 15.58845… and ½ × 6 × 6 × sin 60° = 15.58845…. ✓
Watch out. Use the exact value of sin 60° and keep the surd.
Key words. In triangle ABC the side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. The included angle is the angle between two given sides.
Method. Put the area into Area = ½ ab sin C, rearrange to find sin C, then use the inverse sine.
- Write the area formula and put the numbers in. The area is ½ × a × b × sin C, and here the angle C is unknown.24 = ½ × 8 × 10 × sin C
- Work out ½ × a × b. This is a plain number, so we can combine it.½ × 8 × 10 = 40
24 = 40 sin C - Divide both sides by 40. This gets sin C on its own.sin C = 2440 = 0.6
- Use the inverse sine. The inverse button turns the sine back into an angle.C = sin−1(0.6) = 36.86989…°
- Think about a second (obtuse) angle. Sine of an obtuse angle equals sine of 180° minus it, so both give the same area.Second angle = 180° − 36.86989…° = 143.13010…°
- Round to 1 d.p. Both angles give the same area, so both are answers.C = 36.9° or 143.1°
Check. Work forwards: ½ × 8 × 10 × sin 36.9° = 24.0 cm2, close to 24. ✓
Watch out. The inverse sine only gives the acute angle. An obtuse angle gives the same sine, so check whether the question wants it.