Area

From: James Pallett
Date: 12 March 1999

Subject: Area of a Pentagon

I am wondering how to find the area of a Regular Pentagon. I need to be able to do this for my GCSE coursework on areas of shapes with fixed perimeters.

Maths Help suggests:

When finding areas, we suggest you break your shape into 'easy' shapes:

Rectangles;
Triangles;
Circles;
and any other shapes whose area you can calculate.

Suppose the pentagon has each side of length 10 cm.
In this case, you can split the regular pentagon into 5 identical triangles.
The area of the pentagon is five times the area of one triangle:

The angle A is 360° ÷ 5 = 72°.
Angle B is half of that, 36°.
In the right-angled triangle that includes angle B, 'opp'=5cm and 'adj'=h:
Area of each of the five 'green' triangles is:
So the area of the pentagon is 5×34.4 = 172cm²

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