AREA OF REGULAR HEXAGON- MINDSPARK

What is a Hexagon?

A hexagon is a six-sided polygon. We call a hexagon a regular hexagon when all the six sides and angles are equal. The measure of every angle in a regular hexagon is 120°.

Area Of Regular Hexagon

Area of regular hexagon means the measure of the surface covered by the hexagon. The formula for the area of a regular hexagon is\frac{3 \sqrt{3} a^{2}}{2}where ‘a’ is the side of the regular hexagon.

DERIVATION:

Take a regular hexagon with side ‘a’. Join all the vertices  which are opposite to each other like shown in the figure below:

Now, when we join the vertices opposite to each other, 6 triangles are formed of equal sides and angles.

As we can see in Δ AOB, each angle is 60°.

Therefore we know when in a triangle all the angles are equal, it is known as an equilateral triangle.

We know that Formula of Area of  Equilateral Triangle =\frac{\sqrt{3} a^{2}}{4}where ‘a’ is the side of the equilateral triangle.

Area of regular hexagon

= Area of  Δ AOB +  Area of  Δ BOC + Area of  Δ COD + Area of  Δ DOE + Area of  Δ EOF + Area of  Δ FOA

= \frac{\sqrt{3} a^{2}}{4}+\frac{\sqrt{3} a^{2}}{4}+\frac{\sqrt{3} a^{2}}{4}+\frac{\sqrt{3} a^{2}}{4}+\frac{\sqrt{3} a^{2}}{4}+\frac{\sqrt{3} a^{2}}{4}

= 6 \times \frac{\sqrt{3} a^{2}}{4}

Area of regular hexagon =\frac{3 \sqrt{3} a^{2}}{2}, where a is the side of the regular hexagon.

Examples

1. Find the area of a hexagon whose each side measures 8 \sqrt{2} \mathrm{~cm}?

Solution:

Area of regular hexagon=\frac{3 \sqrt{3} a^{2}}{2}

where ‘a’ is the side of the regular hexagon.

Area of the given regular hexagon =\frac{3 \sqrt{3} \times(8 \times \sqrt{2})^{2}}{2}=\frac{3 \sqrt{3} \times 64 \times 2}{2}=192 \sqrt{3} \mathrm{~cm}^{2}

2. Find the side of the regular hexagon whose area is 150 \sqrt{3} \mathrm{~cm}^{2}

Solution:

Area of regular hexagon =\frac{3 \sqrt{3} a^{2}}{2}

∴ 150 \sqrt{3}=\frac{3 \sqrt{3} a^{2}}{2}

⇒ 50 \times 2=a^{2}
⇒ 100=a^{2}
⇒ a=10

Therefore the length of the side of a regular hexagon will be 10 cm.