Area of Pentagon

by Daisy, Jun 30 2023

What is Pentagon

A pentagon is a shape with 5 sides, like a super cool star! There are different types of pentagons, such as convex and concave ones. A convex pentagon is like a happy face, with all its points facing outward, while a concave pentagon is like a sneaky ninja star, with one or more points pointing inward.

If a convex pentagon has 5 equal sides and its angles are all the same, it's called a regular pentagon. It has special angles: the inside angles are 108°, and the outside angles are 72°. An irregular pentagon is a bit more wild and free, with sides and angles that might not match the others.

The Area of a Pentagon

Now, let's talk about the area of a pentagon, which is like the space it takes up on a piece of paper. To help us understand this, we need to learn some mathematical words and their definitions first:

The "apothem" is a line that goes from the centre of the pentagon to the middle of one side, standing up straight.
The "radius" is a line that reaches from the centre to a pointy corner of the pentagon.
A "side" is the length of one side of the pentagon, and the "perimeter" is the total length of all the pentagon's sides.

For a regular pentagon with side length and apothem, we can use this superpower formula to find its area:

Area of regular pentagon = 1/2 × side length × apothem × 5

And that's how we figure out the area of a pentagon. Now, you can try the question below and find the area of a pentagon:

Q1: Rabbit Bunny discovers that each side of a regular pentagon flower bed is 4 metres long, and which apothem is 3 metres long. What is the area of this pentagon flower bed?

(The Answer will be put at the end of the article.)

Fun Extension

You can also think of a pentagon like a pizza cut into 5 equal slices. Each slice is like a triangle, and they all fit together to make the whole pentagon shape. In this case, the apothem turns into the height of each triangle, and the side length becomes the base.

The space that each triangle takes up, or its area, is found by using this special formula:

Area of triangle = 1/2 × base × height.

Now, since we have 5 delicious triangle-slices, we just multiply the area of one triangle by 5 to find the total area of the whole pentagon. That's what our pentagon formula simplifies to!

But remember, not all pentagons are like a perfect pizza with super simple measurements. So, it's helpful to learn different ways to find the area of a pentagon, just in case you come across a more tricky shape.