Level 7 - NCEA Level 2

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 ×sinθ.

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

$Area=\frac{1}{2}ab\sin C$Area=12absinC

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.

Question 1

Calculate the area of the following triangle.

Round your answer to two decimal places.

A triangle is depicted with the measurements of its two sides and their included angle. The included angle, highlighted by a blue-shaded arc, measures $44^\circ$44° and is adjacent to sides measuring $3$3 m and $5.7$5.7 m.

Question 2

Calculate the area of the triangle.

Round your answer to two decimal places.

A triangle is illustrated given the measure of one of its interior angle. The angle on the upperpart of the triangle measures $123^\circ$123°, indicated by an arc shaded with blue. The shorter adjacent side of the $123^\circ$123°-degree angle measures $4$4 m while the longer adjacent side of the $123^\circ$123°-degree angle measures $7.4$7.4 m.

Question 3

Calculate the area of the following triangle.

Round your answer to the nearest square centimetre.

A non-right-angled triangle with vertices labeled $P$`P`, $Q$`Q`, and $R$`R`. Angle $QPR$`QPR` measures $39$ degrees, while angle $PQR$`PQR` measures $25$ degrees. Side $QR$`QR`, which is opposite angle $QPR$`QPR` measures $33$ $cm$`cm`. Side $PR$`PR`, which is opposite angle $PQR$`PQR` measures $22$ $cm$`cm`.

Outcomes

M7-4

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

91259

Apply trigonometric relationships in solving problems