Sheet № 218 · Higher only · AQA · Edexcel · OCR
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
§Key definitions
Question:
Find the area of a triangle with sides 8 cm and 11 cm and an included angle of 40°.
Answer:
The area is 28.3 cm².
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.
§Formulas to memorise
Area = ½ab sin C
Area = ½bc sin A
Area = ½ac sin B
Substitute into Area = ½ × a × b × sin C.
Worked example
Find the area of a triangle with sides 8 cm and 11 cm and an included angle of 40°.
This topic is Higher only, but this example uses straightforward values.
⚠ 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.