Despite having one more side/edge than a 5 sided Pentagon.
Trying to calculate the area of a Hexagon with 6 sides is generally simpler to establish the area of.
Derive Formula for Area of Hexagon
If we look at a regular Hexagon shape with 6 sides, all of which are the same length.
A regular Hexagon such as this can be divided into 6 equilateral triangles, by drawing straight lines from the central point of the Hexagon to each corner.
Each of these triangles will be the same size.
In order to calculate the area of the Hexagon in total, we would need to work out the area of one of the inner triangles, and then multiply this value by 6.
Area of an Inner Triangle:
Provided that we know the length of a side of a hexagon a.The area of one of the equilateral triangles is given by: \frac{\sqrt3 a^2}{4}
As seen, there are 6 equilateral triangles inside a regular Hexagon, so we can multiply by 6.
6 × \frac{\sqrt3 a^2}{4} = \frac{6\sqrt3 a^2}{4} = \frac{3\sqrt3 a^2}{2}
The result is the formula for the area of a Hexagon, which we can use to find the area of a regular Hexagon, provided we know the length of the sides.
Examples
1.1
Find the area of the following Hexagon.
Solution
Area = \frac{3\sqrt3 (6)^2}{2} = \frac{3\sqrt3 \times 36}{2} = \frac{108\sqrt3}{2} = 93.53
Hexagon Area is 93.53cm2.
1.2
Find the area of the following Hexagon.
Solution
Here it’s the diameter of the Hexagon that we know, instead of the side length.
But that is still helpful, as the length of the diameter of a regular Hexagon is double the length of a side.
So dividing 8 by 2 will give the value of the length of a side of the Hexagon.
8 ÷ 2 = 4
The length of a side of the Hexagon is 5cm.
Area = \frac{3\sqrt3(4)^2}{2} = \frac{3\sqrt3 \times 16}{2} = \frac{48\sqrt3}{2} = 41.57
Hexagon Area is 41.57cm2.
Calculate Area of a Hexagon
Alternative Method
There is also another approach that can be used to calculate the area of a Hexagon.
If you know the length of an apothem of a regular Hexagon as well as the length of all sides combined.
Then Hexagon Area can be found by: \frac{1}{2} × Apothem × Perimeter
The apothem in a Hexagon is a straight line from the centre of a side to the centre of the hexagon itself, labelled b below.
The size of an apothem line in a regular polygon such as a regular Hexagon is given by the following.
Apothem Line = \bf{\frac{a}{2 \space \times \space tan(\frac{180}{n})}}
Where n is the number of sides, and a is the length of one side.
Examples
2.1
Find the area of the following Hexagon.
Solution
Apothem line = \bf{\frac{a}{2 \space \times \space tan(\frac{180}{n})}} ( This can help find a. )
=> 4 = \bf{\frac{a}{2 \space \times \space tan(\frac{180}{6})}} = \bf{\frac{a}{2 \space \times \space tan(30)}} = \bf{\frac{a}{\frac{2}{\sqrt3}}}
=> 4 = \bf{\frac{a}{\frac{2}{\sqrt3}}}
=> 4 × \bf{\frac{2}{\sqrt3}} = a , \bf{\frac{8}{\sqrt3}} = a
a = 4.62
HEXAGON PERIMETER = 4.62 × 6 = 27.72
Hexagon Area = \frac{1}{2} × 27.72 × 4 = 55.44
The area of the Hexagon is 55.44cm2.