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Trigonometry
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United KingdomEngland
Key Stage 4

Area of Non-Right Angled Triangles

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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=12​absinC

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.

 

 

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