Trigonometry

NZ Level 7 (NZC) Level 2 (NCEA)

Area of Non-Right Angled Triangles

Lesson

We know how to find the area of right-angled triangles, but how can we find the area of a non right-angled triangle?

If we know two sides and the included angle (SAS), there is another formula we can use to find the area.

$\text{Area }=\frac{1}{2}\times\text{Side 1 }\times\text{Side 2 }\times\sin\theta$Area =12×Side 1 ×Side 2 ×`s``i``n``θ`.

It really doesn't matter what you call the sides as long as you have two sides and the included angle. It's worth noting that we always label the sides with lower case letters, and the angles directly opposite the sides with a capital of the same letter. The formula is most commonly written as follows:

Area rule

$Area=\frac{1}{2}ab\sin C$`A``r``e``a`=12`a``b``s``i``n``C`

Where $a$`a` and $b$`b` are the known side lengths, and $C$`C` is the given angle between them, as per the diagram above.

Let's have a look at some worked examples.

Calculate the area of the following triangle.

Round your answer to two decimal places.

Calculate the area of the triangle.

Round your answer to two decimal places.

Calculate the area of the following triangle.

Round your answer to the nearest square centimetre.

Apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions

Apply trigonometric relationships in solving problems