Area of a Triangle Using ½ab sin C
Grade 6The formula
The standard area formula ½ × base × height only works easily when you have a perpendicular height. The sine area formula bypasses that — it uses two sides and the included angle directly.
Here a and b are two sides of the triangle, and C is the angle between them.
How it connects to the standard formula
The perpendicular height h = a sin C (trig in a right-angled triangle). Substitute into ½ × b × h and you get ½ab sin C. So the formula is really the same idea — trig gives you the height you couldn't measure directly.
Worked example
Try this one
A student needs to find a missing side in a triangle.
They know two sides and the angle between them.
(a) Which rule should they use: the sine rule or the cosine rule? [1]
(b) Write down the cosine rule for finding side \( a \). [1]
This tests when each rule applies.
Ask yourself: 'do I have a matching side and its opposite angle (sine rule), or two sides and the angle between them (cosine rule)?' Think about which pieces of information you are given.
