$\text{Area }=\frac{1}{2}\times\text{Side 1 }\times\text{Side 2 }\times\sin\text{angle between the sides }$Area =12×Side 1 ×Side 2 ×sinangle between the sides .

This formula can be written using $3$3 combinations of sides a,b and c. 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. This is a standard in triangle notation.

Area rule

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

Herons Formula

Herons formula is a special formula used to find the area of triangles if all we know is the lengths of the 3 sides. It is called Herons Formula after Hero of Alexandria. Hero of Alexandria was a Greek Engineer and Mathematician in 10 – 70 AD, he is also attributed to the development of the world's first steam engine, even if it was just considered a toy at the time.

Herons Formula

$A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$A=√s(s−a)(s−b)(s−c)

where $a$a, $b$b and $c$c are the lengths of the $3$3 sides and $s$s is the value for half the perimeter, ie $s=\frac{1}{2}\left(a+b+c\right)$s=12(a+b+c)

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`.

Area of Non Right Angle Triangles

Area of a Triangle rule

Herons Formula

Worked Examples

Question 1

Question 2

Question 3

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