NZ Level 7 (NZC) Level 2 (NCEA)
Area of non right angled triangles
Lesson

## Area of a Triangle rule

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

$\text{Area }=\frac{1}{2}\times\text{Side 1 }\times\text{Side 2 }\times\sin\text{angle between the sides }$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(sa)(sb)(sc)

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.

##### Question 2

Calculate the area of the triangle.

##### Question 3

Calculate the area of the following triangle.

### Outcomes

#### M7-4

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

#### 91259

Apply trigonometric relationships in solving problems