Area of a Triangle (HSC SSCE Mathematics Standard): Revision Notes

Area of a Triangle

Introduction

When working with triangles that don't have a right angle, we can still calculate their area using a special formula. To use this method, you need to know two sides of the triangle and the angle between them (called the included angle).

The key principle is: the area of a triangle equals half of the product of two sides multiplied by the sine of the angle between those sides.

Deriving the formula

To understand where this formula comes from, we start by drawing a perpendicular line from one vertex to the opposite side. This creates a right-angled triangle within our original triangle.

Starting with the basic area formula:

$A = \frac{1}{2}bh$

where $h$ is the perpendicular height.

In the right-angled triangle $\triangle BCD$, we can use the sine ratio:

$\sin C = \frac{h}{a}$

Rearranging this gives us:

$h = a \sin C$

Now we substitute this expression for $h$ into the area formula:

$A = \frac{1}{2}b \times a \sin C$

$A = \frac{1}{2}ab \sin C$

The three area formulas

There are three equivalent ways to express the area formula, depending on which two sides and included angle you know:

$A = \frac{1}{2}bc \sin A$

$A = \frac{1}{2}ac \sin B$

$A = \frac{1}{2}ab \sin C$

Using the formula - a worked example

Exam tips

Remember!