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Area and perimeter: Formulae

Area and perimeter: Formulae

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Summary

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

Area=12×base×perpendicular height\text{Area}=\frac{1}{2}\times\text{base}\times\text{perpendicular\,height}​​


A=12bhA=\frac{1}{2}bh​​

Maths; Shapes and area; KS4 Year 10; Area and perimeter: Formulae
bb​​

Square

Area=length2\text{Area}=\text{length}^2​​


A=x2A=x^2​​

​​

Maths; Shapes and area; KS4 Year 10; Area and perimeter: Formulae
xx​​

Rectangle

Area=length×width\text{Area}=\text{length}\times\text{width}​​


A=lwA=lw​​

Maths; Shapes and area; KS4 Year 10; Area and perimeter: Formulae

Parallelogram

Area=base×perpendicular height\text{Area}=\text{base}\times\text{perpendicular\,height}​​


A=bhA=bh​​

Maths; Shapes and area; KS4 Year 10; Area and perimeter: Formulae

Trapezium

Area=12×(a+b)×height\text{Area}=\frac{1}{2}\times(a+b)\times\text{height},


A=12(a+b)×hA=\frac{1}{2}(a+b)\times h​​


where aa and bb are the lengths of the two parallel sides.

Maths; Shapes and area; KS4 Year 10; Area and perimeter: Formulae


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=4ma=5m,b=9m,c=4m​ and h=3mh=3m. What is the area and exact perimeter of the shape?

Maths; Shapes and area; KS4 Year 10; Area and perimeter: Formulae

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:

12×base×perpendicular height=12×b×c=12×9×4\frac{1}{2}\times\text{base}\times\text{perpendicular\,height}=\frac{1}{2}\times b\times c=\frac{1}{2}\times9\times4​​


Atriangle=18m2A_{triangle}=18m^2​​


The area of the parallelogram is given to be:

base×perpendicular height=b×h=9×3\text{base}\times\text{perpendicular\,height}=b\times h=9\times3​​


Aparallelogram=27m2A_{parallelogram}=27m^2​​


Add these two areas together to find the total area:

A=Atriangle+Aparallelogram=18+27A=A_{triangle}+A_{parallelogram}=18+27​​


A=45m2A=45m^2​​


The are of the shape is 45m2\underline{45m^2}.


Perimeter:

The perimeter of the shape is given to be:

c+a+b+a+d=23+dc+a+b+a+d=23+d


Where dd is the hypotenuse of the right-angled triangle.

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

b2+c2=d2b^2+c^2=d^2​​

92+42=d29^2+4^2=d^2​​

d2=81+16=97d^2=81+16=97​​

d=97md=\sqrt{97}m​​


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


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FAQs - Frequently Asked Questions

What is the formula for the area of a parallelogram?

What is the formula for the area of a trapezium?

What is the formula for the area of a triangle?

What is perimeter?

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