© T Madas

© T Madas

There is a trigonometric formula for the area of a triangle. This is how it is derived:

b

a h

θ

sinθ =OppHyp

sinθ =ha

h = ax sinθ

A = 12 b´ h´ Û

A =12 b´ sina ´ ´ Û

12 sinA ab =

12 sinA ab =

© T Madas

Find the area of this triangle:

8 cm

6 cm

30°

A =12 6´ 8´ Û

212 cmA =

12 sinA ab =

sin30°´

© T Madas

© T Madas

8 cm

Calculate the area of a regular hexagon of side 8 cm, giving your answer to 3 significant figures.

A B

C

DE

FO

60°

60°

8 cm

60°

AT = 12

x 8x 8 x sin60°

AT =32 sin60°

AH =192 sin60°

AH = 166 cm2 [ 3 s.f.]

© T Madas

© T Madas

A regular decagon is inscribed in a circle of radius 4 cm.Calculate the area of the octagon, giving your answer correct to 3 significant figures.

4 cm

36°O

A

B

x 4A = x 4 x sin36°

The area of the triangle OAB

A ≈4.702 cm2

The area of the decagon

10x 4.702= 47.0 cm2 [ 3 s.f.]

12

© T Madas

© T Madas

40 m

The triangle ABC has AB = 30 m, ABC = 30° and has an area of 300 m2.Calculate the perimeter of the triangle, giving your answer correct to 3 significant figures.

A

B

C

30°

30 m

300 m2

x xA = x 30

x sin30°

The area of the triangle ABC

12

x

y

x x300 = x 30

x sin30° 12

x x300 = x 30

x 12

12

x 4

4 x

1200 = 30x

x = 40

© T Madas

20.53 m

The triangle ABC has AB = 30 m, ABC = 30° and has an area of 300 m2.Calculate the perimeter of the triangle, giving your answer correct to 3 significant figures.

A

B

C

30°

30 m

300 m2

y

– 2400

x 30y 2 = 302+402– 2 x 40x cos30°

By the cosine rule on ABC

y 2 = 900+ 1600 cos30°

y 2 ≈421.539

y ≈20.53 m

40 m

Therefore the perimeter of ABC to 3 s.f. is90.5 m

© T Madas