Area Using ½ab sin C –

The formula Area = ½ab sin C lets you find the area of any triangle when you know two sides and the included angle. It is a Higher-tier GCSE Maths topic tested on AQA, Edexcel, and OCR papers, and it frequently appears alongside the sine and cosine rules in multi-step problems. This guide explains the formula, shows when and how to use it, and provides worked examples and practice questions.

What Is the ½ab sin C Formula?

For any triangle with sides a, b, and c opposite angles A, B, and C respectively:

Key Formula

Here, a and b are any two sides, and C is the angle between those two sides (the included angle). You can also write this as:

• Area = ½bc sin A
• Area = ½ac sin B

All three forms give the same result — choose the one that uses the two sides and included angle you know.

When to Use This Formula

Use ½ab sin C when you have:

• Two sides and the included angle (the angle between the two known sides).
• It works for any triangle — not just right-angled triangles.

If you do not have the included angle, you may need the sine or cosine rule first to find it.

Step-by-Step Method

1. Identify two sides and the angle between them.
2. Label the sides a and b, and the included angle C.
3. Substitute into Area = ½ × a × b × sin C.
4. Evaluate sin C using your calculator (ensure it is set to degrees).
5. Multiply to find the area.

Worked Example 1 — Foundation Level

This topic is Higher only, but this example uses straightforward values.

Question: Find the area of a triangle with sides 8 cm and 11 cm and an included angle of 40°.

Working:

Step 1 — a = 8, b = 11, C = 40°.

Step 2 — Area = ½ × 8 × 11 × sin 40°.

Step 3 — sin 40° = 0.6428 (4 d.p.).

Step 4 — Area = ½ × 8 × 11 × 0.6428 = 0.5 × 56.565 = 28.3 cm² (1 d.p.).

Answer: The area is 28.3 cm².

Worked Example 2 — Higher Level

Question: Triangle PQR has PQ = 14 cm, PR = 9 cm, and area = 50.4 cm². Find angle P.

Working:

Step 1 — Area = ½ × PQ × PR × sin P.

Step 2 — 50.4 = ½ × 14 × 9 × sin P = 63 sin P.

Step 3 — sin P = 50.4 / 63 = 0.8.

Step 4 — P = sin⁻¹(0.8) = 53.1° (1 d.p.).

Answer: Angle P = 53.1°.

Worked Example 3 — Exam Style

Question: In triangle ABC, AB = 10 cm, BC = 7 cm, and angle B = 65°. Find the area of the triangle and then use the cosine rule to find AC.

Working:

Part 1 — Area:

Step 1 — a = BC = 7, c = AB = 10, included angle B = 65°.

Step 2 — Area = ½ × 7 × 10 × sin 65° = 35 × 0.9063 = 31.7 cm² (1 d.p.).

Part 2 — Finding AC using the cosine rule:

Step 3 — AC² = AB² + BC² − 2(AB)(BC)cos B = 100 + 49 − 2(10)(7)cos 65°.

Step 4 — AC² = 149 − 140 × 0.4226 = 149 − 59.164 = 89.836.

Step 5 — AC = √89.836 = 9.5 cm (1 d.p.).

Answer: Area = 31.7 cm² and AC = 9.5 cm.

Common Mistakes

• Using the wrong angle. The angle must be the one between the two sides you are using. If you use an angle opposite one of the sides, you will get the wrong answer.
• Calculator in radians mode. Make sure your calculator is set to degrees, not radians. Sin 40 in radians gives a completely different value.
• Forgetting the ½. The formula includes a factor of ½ — without it your answer will be double the correct area.

Exam Tips

• The formula Area = ½ab sin C is provided on the exam formula sheet, so you do not need to memorise it — but you do need to know when and how to use it.
• If the question gives you a non-right-angled triangle and asks for the area, this is almost certainly the formula to use.
• Working backwards (given the area, find the angle or a side) is a common Higher-tier question pattern.
• Combine with the sine or cosine rule when you need to find extra information before using the area formula.

Practice Questions

Q1 (Higher): Find the area of a triangle with sides 12 cm and 15 cm and an included angle of 50°.

Q2 (Higher): A triangle has sides 9 cm and 6 cm with an included angle of 130°. Find the area.

Q3 (Higher): The area of a triangle is 40 cm². Two sides are 10 cm and 12 cm. Find the included angle.

Practise triangle area questions with instant feedback free on GCSEMathsAI.

Related Topics

• Trigonometry: Sine and Cosine Rules — finding sides and angles in non-right triangles.
• Trigonometry: SOHCAHTOA — trigonometry in right-angled triangles.
• Area of a Sector and Segment — using ½ab sin C for segment area.

Summary

Area = ½ab sin C is the go-to formula for finding the area of any triangle when you know two sides and the included angle. The included angle must be between the two known sides. This formula extends triangle area calculations beyond right-angled triangles and is often combined with the sine or cosine rule in multi-step Higher-tier questions. Always check your calculator is in degrees mode, do not forget the factor of ½, and make sure you are using the correct angle.