Question

What is the area of a regular hexagon with apothem $ 7.5 $ inches? What is its perimeter?

Answer 1

Hint: We find the regular hexagon with given height of the equilateral triangles being $ 7.5 $ inches. The height gives the sides of the hexagon which in turn gives area and perimeter following the formula of $ \dfrac{3\sqrt{3}{{a}^{2}}}{2} $ and $ 6a $ respectively.

Complete step by step solution:
A regular hexagon is a closed shape polygon which has six equal sides and six equal angles. In case of any regular polygon, all its sides and angles are equal. For instance, a regular pentagon has 5 equal sides and a regular octagon has 8 equal sides. It is made up of six equilateral triangles.
We need to find the side of the hexagon to find the area and perimeter.
One of the triangles has a height of $ 7.5 $ inches.
The triangle with side $ a $ will have $ \dfrac{a\sqrt{3}}{2} $ as height which gives $ \dfrac{a\sqrt{3}}{2}=7.5 $ .
Therefore, the value of $ a $ is $ a=7.5\times \dfrac{2}{\sqrt{3}}=5\sqrt{3} $ .
We know that area and perimeter of a regular hexagon with side length $ a $ will be $ \dfrac{3\sqrt{3}{{a}^{2}}}{2} $ and $ 6a $ respectively.
Putting the values, we get
 $ \dfrac{3\sqrt{3}{{a}^{2}}}{2}=\dfrac{3\sqrt{3}}{2}{{\left( 5\sqrt{3} \right)}^{2}}=\dfrac{225\sqrt{3}}{2} $ and $ 6a=6\times 5\sqrt{3}=30\sqrt{3} $ .
Therefore, the area and perimeter of the regular hexagon is $ \dfrac{225\sqrt{3}}{2} $ square inches and $ 30\sqrt{3} $ inches.

Note: To find the area we can also find the area of one triangle and multiply with 6 to find the same solution of the problem. The triangles are created from the centre of the figure which gives 6 equilateral triangles.