Find the area of a triangle from two sides and the angle between them, or work backwards to find the angle. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
The area of a triangle is 21absinC when you know two sides and the included angle. Included means the angle sitting between those two sides. If the angle is not included, you cannot use this formula until you have found the included angle (sine rule, maybe). A right-angled triangle can still use 21bh; both should agree.
Leave in terms of sin if the angle is exact (60∘) and a non-calculator paper wants a surd. Calculator: sin of the angle, then multiply, 3 s.f. An obtuse included angle is fine — sin of obtuse is still positive, so the area is positive.
Area =21absinC with C the included angle. sin60∘=23, sin90∘=1. Working backwards: sinC=ab2×area, then sin−1 (acute or obtuse from context).
Objectives
Use 1/2 ab sin C with sin 30.
Use sin 60 and sin 45 in exact form.
Use the right angle case sin 90 = 1.
Key points
Identify the included angle — the angle between the two known sides.
Use Area=21absinC with the calculator in degrees.
Use exact values (sin30∘=21, sin60∘=23) on a non-calculator paper.
Rearrange to find an angle or a side when the area is given.
Worked examples
Example 1
Find the area of the triangle with sides 8 and 10 and included angle 30.
2 marks
State area formula: Area = 1/2 × a × b × sin C.
Substitute sides and sin 30: 1/2 × 8 × 10 × (1/2).
Calculate area: 1/2 × 80 × 1/2 = 20.
Answer: 20
Common slips
Slip: Forget the 1/2: a × b × sin C. Triangle area requires the factor of 1/2.
Slip: Use sin 30 = 1. sin 30° = 1/2.
Slip: Stop at 40. Both factors of 1/2 must be applied.
Example 2
Find the area of the triangle with sides 6 and 6 and included angle 60.
2 marks
Substitute values into formula: Area = 1/2 × 6 × 6 × sin 60.
Use exact value sin 60 = √3/2: 1/2 × 36 × (√3/2).
Simplify: 18 × (√3/2) = 9√3.
Answer: 9√3
Common slips
Slip: Use 6 × 6 = 36 without sine. The included angle sine must be included.
Slip: Replace sin 60 with decimal. Leave answer in exact surd form.
Slip: Write 18. Exact area is 9√3.
Example 3
Find the area of the triangle with sides 6 and 6 and included angle 45.
3 marks
Substitute into formula: Area = 1/2 × 6 × 6 × sin 45.
Use exact value sin 45 = √2/2: 18 × (√2/2).
Simplify: 18/2 × √2 = 9√2.
Answer: 9√2
Common slips
Slip: Add side lengths. Area formula multiplies the two bounding sides.
Slip: Drop √2. sin 45° = √2/2.
Slip: Write 9 by halving 18 incorrectly. Exact area is 9√2.
Questions and answers
GCSE Maths area of any triangle questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Find the area of the triangle with sides 6 and 10 and included angle 30.
Show answer
Answer: 15
On sides 6 and 10, sin 30 = 1/2.
Area = 1/2 × 6 × 10 × 1/2 = 15.
M1 — On sides 6 and 10, sin 30 = 1/2.
A1 — Answer 15
Question 2 Easy2 marks
Find the area of a triangle with sides 8 cm and 11 cm and included angle 37 degrees to 3 significant figures.
Show answer
Answer: 26.5
Use 1/2 ab sin C with a calculator.
The area is 26.5.
M1 — Use 1/2 ab sin C with a calculator.
A1 — Answer 26.5
Question 3 Medium2 marks
The sides are 8 and 10 and the area is 20. What is the acute included angle?
Show answer
Answer: 30
sin C = 2×20 / (8×10) = 1/2.
The acute angle is 30 degrees.
M1 — sin C = 2×20 / (8×10) = 1/2.
A1 — Answer 30
Question 4 Medium2 marks
Sides 10 cm and 10 cm and area 30. What is the acute included angle to the nearest degree?
Show answer
Answer: 37
sin C = 60/100 = 0.6.
The acute angle is 37 degrees.
M1 — sin C = 60/100 = 0.6.
A1 — Answer 37
Question 5 Hard2 marks
The area of a triangle is 36 and the perpendicular height is 6 cm. What is the base?
Show answer
Answer: 12
Area = 1/2 × base × height.
Base = 2 × 36 / 6.
M1 — Area = 1/2 × base × height.
A1 — Answer 12
Question 6 Hard2 marks
Find the area of a triangle with sides 7 cm and 9 cm and included angle 100 degrees to 3 significant figures. The included angle is obtuse.
Show answer
Answer: 31.0
sin 100 = sin 80.
1/2 × 7 × 9 × sin 100 = 31.0 to 3 significant figures.
M1 — sin 100 = sin 80.
A1 — Answer 31.0
Question 7 Exam-style4 marks
Find the area of the triangle with sides 5 and 8 and included angle 60. Give the area as a surd.
Show answer
Answer: 10√3
sin 60 = √3/2.
The area is 10√3.
M1 — sin 60 = √3/2.
M1 — The area is 10√3.
A1 — The area is 10√3.
B1 — Answer 10√3
Common mistakes
Most lost marks on this topic are one slip.
Forgetting sin of the included angle
Multiply by sin C as well as the half
Replacing √3/2 with 1
Keep the exact sine
Using an angle that is not between the two sides
The angle in the formula is the included one
Spot the slip
The area formula uses sin of the angle between the two sides. Which angle is that?
The included angle
Any angle
The smallest angle
Show the slip
The included angle The two sides must enclose the angle in the formula.
Exam tips
A calculator is useful here. Show the calculation you typed, not only the final line.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the formula with the numbers substituted before pressing anything — that line is the method mark.
On the non-calculator paper, angles of 30∘, 45∘ or 60∘ mean exact values are expected; leave surds in the answer.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.