M
Math Coach Amy
Amy Ferguson Moncure

IB AA · Unit 2 · 5B Rules of Trigonometry

Area of a Triangle

Find a triangle’s area from two sides and the included angle, or work backwards to an angle or a side.

Jump to section:

Key Ideas Easy Medium Hard

Area = 12 ab sin C

What to Remember

Area from two sides and the included angle

Area = 12 ab sin C

Angle C sits between sides a and b.

Find the angle

sin C = 2 × areaa b

These problems ask for the acute angle.

Find a side

a = 2 × areab sin C

Easy

Find the area

The angle in the formula is the included angle.

Example
In triangle ABC, a = 10 cm, b = 16 cm, and angle C = 42°. Find the area.

Area = 12 a b sin C

Area = 12 × 10 × 16 × sin 42°

Area = 53.5 cm²

Answer: 53.5 cm²

1
In triangle ABC, a = 8 cm, b = 13 cm, and angle C = 37°. Find the area.
Show Answer

Area = 12 a b sin C

Area = 12 × 8 × 13 × sin 37°

Area = 31.3 cm²

Answer: 31.3 cm²

2
In triangle ABC, a = 12 cm, b = 9 cm, and angle C = 65°. Find the area.
Show Answer

Area = 12 a b sin C

Area = 12 × 12 × 9 × sin 65°

Area = 48.9 cm²

Answer: 48.9 cm²

3
In triangle ABC, a = 7 cm, b = 7 cm, and angle C = 50°. Find the area.
Show Answer

Area = 12 a b sin C

Area = 12 × 7 × 7 × sin 50°

Area = 18.8 cm²

Answer: 18.8 cm²

4
In triangle ABC, a = 15 cm, b = 11 cm, and angle C = 28°. Find the area.
Show Answer

Area = 12 a b sin C

Area = 12 × 15 × 11 × sin 28°

Area = 38.7 cm²

Answer: 38.7 cm²

5
In triangle ABC, a = 6.40 cm, b = 9.10 cm, and angle C = 73°. Find the area.
Show Answer

Area = 12 a b sin C

Area = 12 × 6.40 × 9.10 × sin 73°

Area = 27.8 cm²

Answer: 27.8 cm²

Medium

Find the acute angle

Rearrange for sin C, then use the inverse sine.

Example
In triangle ABC, a = 12 cm, b = 9 cm, and the area is 30 cm². Find acute angle C.

Area = 12 a b sin C

30 = 12 × 12 × 9 × sin C

sin C = 6012 × 9

sin C = 0.556

C = 33.7°

Answer: 33.7°

1
In triangle ABC, a = 10 cm, b = 8 cm, and the area is 28 cm². Find acute angle C.
Show Answer

Area = 12 a b sin C

28 = 12 × 10 × 8 × sin C

sin C = 5610 × 8

sin C = 0.700

C = 44.4°

Answer: 44.4°

2
In triangle ABC, a = 14 cm, b = 11 cm, and the area is 40 cm². Find acute angle C.
Show Answer

Area = 12 a b sin C

40 = 12 × 14 × 11 × sin C

sin C = 8014 × 11

sin C = 0.519

C = 31.3°

Answer: 31.3°

3
In triangle ABC, a = 6 cm, b = 9 cm, and the area is 15 cm². Find acute angle C.
Show Answer

Area = 12 a b sin C

15 = 12 × 6 × 9 × sin C

sin C = 306 × 9

sin C = 0.556

C = 33.7°

Answer: 33.7°

4
In triangle ABC, a = 13 cm, b = 7 cm, and the area is 22 cm². Find acute angle C.
Show Answer

Area = 12 a b sin C

22 = 12 × 13 × 7 × sin C

sin C = 4413 × 7

sin C = 0.484

C = 28.9°

Answer: 28.9°

5
In triangle ABC, a = 16 cm, b = 12 cm, and the area is 50 cm². Find acute angle C.
Show Answer

Area = 12 a b sin C

50 = 12 × 16 × 12 × sin C

sin C = 10016 × 12

sin C = 0.521

C = 31.4°

Answer: 31.4°

Hard

Find the missing side

Rearrange so the unknown side is the result.

Example
In triangle ABC, b = 10 cm, angle C = 35°, and the area is 48 cm². Find side a.

Area = 12 a b sin C

48 = 12 × a × 10 × sin 35°

a = 9610 sin 35°

a = 16.7 cm

Answer: 16.7 cm

1
In triangle ABC, one side is 8 cm, the included angle A is 52°, and the area is 36 cm². Find side b.
Show Answer

Area = 12 × b × 8 × sin A

36 = 12 × b × 8 × sin 52°

b = 728 sin 52°

b = 11.4 cm

Answer: 11.4 cm

2
In triangle ABC, one side is 11 cm, the included angle C is 28°, and the area is 20 cm². Find side a.
Show Answer

Area = 12 × a × 11 × sin C

20 = 12 × a × 11 × sin 28°

a = 4011 sin 28°

a = 7.75 cm

Answer: 7.75 cm

3
In triangle ABC, a = 9 cm, b = 14 cm, and angle C = 47°. Find side c and the area.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (9)² + (14)² − 2 (9) (14) cos 47°

c = 10.3 cm

Area = 12 × 9 × 14 × sin 47°

Area = 46.1 cm²

Answer: c = 10.3 cm, area = 46.1 cm²

4
In triangle ABC, a = 13 cm, angle B = 100°, and the area is 40 cm². Find side c.
Show Answer

Area = 12 a c sin B

40 = 12 × 13 × c × sin 100°

c = 8013 sin 100°

c = 6.25 cm

Answer: 6.25 cm

5
The area of triangle ABC is 75 cm². Angle B = 64° and side a = 12 cm. Find side c.
Show Answer

Area = 12 a c sin B

75 = 12 × 12 × c × sin 64°

c = 15012 sin 64°

c = 13.9 cm

Answer: 13.9 cm

Want to go over a solid or a triangle together?

Book a Session with Amy