Area and perimeter: Formulae

​​In a nutshell

The perimeter and area of triangles and quadrilaterals can be calculated with their own formula. These formulae can also be applied to finding the area and perimeter of compound shapes.

Area and perimeter - definitions

The perimeter of a shape is the sum of the lengths of the outer sides of the shape.

The area of a shape tells you the size of the surface of the shape.

Area formulae

These are the following formulae you need to know.

​​shape	formula	illustration
Triangle	$$\text{Area}=\frac{1}{2}\times\text{base}\times\text{perpendicular\,height}$$​​ ​$$A=\frac{1}{2}bh$$​​	​ $$b$$​​
Square	$$\text{Area}=\text{length}^2$$​​ ​$$A=x^2$$​​	​​ ​$$x$$​​
Rectangle	$$\text{Area}=\text{length}\times\text{width}$$​​ ​$$A=lw$$​​
Parallelogram	$$\text{Area}=\text{base}\times\text{perpendicular\,height}$$​​ ​$$A=bh$$​​	​
Trapezium	$$\text{Area}=\frac{1}{2}\times(a+b)\times\text{height}$$, $$A=\frac{1}{2}(a+b)\times h$$​​ where $$a$$ and $$b$$ are the lengths of the two parallel sides.

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Area and perimeter problems

You can use the formulae for the areas and perimeters of triangles and quadrilaterals to solve problems involving more complex shapes.

Example 1

In the diagram below, let $$a=5m,b=9m,c=4m$$​ and $$h=3m$$. What is the area and exact perimeter of the shape?

Area:

The area of the shape is the area of the parallelogram $$+$$ the area of the triangle.

The area of the triangle is given to be:

$$\frac{1}{2}\times\text{base}\times\text{perpendicular\,height}=\frac{1}{2}\times b\times c=\frac{1}{2}\times9\times4$$​​

$$A_{triangle}=18m^2$$​​

The area of the parallelogram is given to be:

$$\text{base}\times\text{perpendicular\,height}=b\times h=9\times3$$​​

​$$A_{parallelogram}=27m^2$$​​

Add these two areas together to find the total area:

$$A=A_{triangle}+A_{parallelogram}=18+27$$​​

​$$A=45m^2$$​​

The are of the shape is $$\underline{45m^2}$$.

Perimeter:

The perimeter of the shape is given to be:

​$$c+a+b+a+d=23+d$$

Where $$d$$ is the hypotenuse of the right-angled triangle.

The length of the hypotenuse can be found by using Pythagoras' theorem:

​$$b^2+c^2=d^2$$​​

​$$9^2+4^2=d^2$$​​

​$$d^2=81+16=97$$​​

​$$d=\sqrt{97}m$$​​

The exact perimeter is $$\underline { (23+\sqrt{97}) \ m}$$.​​