Everything Maths Grade 11

17.5 CHAPTER 17. TRIGONOMETRY

The Area Rule EMBDP

DEFINITION: The Area Rule

The area rule applies toany triangle and states that the area of a triangle is given by

half the product of any two sides with the sine of the angle between them.

That means that in the�DEF, the area is given by:

A =

1

2

DE. EF sinEˆ

=

1

2

EF. F DsinFˆ

=

1

2

F D. DE sinDˆ

�

D

�

E

�

F

In order show that this is true for all triangles, consider�ABC.

� �

A

�

B

�

C

b h a

c

The area of any triangleis half the product of thebase and the perpendicular height. For�ABC, this
is:

A =

1

2

c. h.

However, h can be written in termsofAˆ as:

h = bsinAˆ

So, the area of�ABC is:

A =

1

2

c. bsinA.ˆ

Using an identical method, the area rule can beshown for the other twoangles.