Suppose a circle of radius r is inscribed in a hexagon. What is the area of the hexagon?

Question

Evelyn Arboleda · Accepted Answer

Area of a regular hexagon with a radius of inscribed circle #r# is
#S=2sqrt(3)r^2#

Obviously, a regular hexagon can be considered as consisting of six equilateral triangles with one common vertex at the center of an inscribed circle.

The altitude of each of these triangles equals to #r#.

The base of each of these triangles (a side of a hexagon that is perpendicular to an altitude-radius) equals to
#r*2/sqrt(3)#

Therefore, an area of one such triangle equals to
#(1/2)*(r*2/sqrt(3))*r=r^2/sqrt(3)#

The area of an entire hexagon is six times greater:
#S = (6r^2)/sqrt(3) = 2sqrt(3)r^2 #

Autumn Armstrong · Answer

undefined The area of the hexagon can be found using the formula:

$$ \text{Area of Hexagon} = 6 \times \text{Area of Triangle} $$

Where each triangle's area can be calculated as half the product of the apothem (radius of the inscribed circle) and the side of the hexagon.

$$ \text{Area of Triangle} = \frac{1}{2} \times \text{apothem} \times \text{side} $$

In this case, the apothem of the hexagon (radius of the inscribed circle) is also the height of each triangle. For a regular hexagon inscribed in a circle, the radius of the circle is equal to the length of each side of the hexagon.

Therefore, the area of each triangle is:

$$ \text{Area of Triangle} = \frac{1}{2} \times r \times r $$

$$ \text{Area of Triangle} = \frac{1}{2} r^2 $$

Then, the area of the hexagon becomes:

$$ \text{Area of Hexagon} = 6 \times \left(\frac{1}{2} r^2\right) $$

$$ \text{Area of Hexagon} = 3r^2 $$

So, the area of a regular hexagon inscribed in a circle of radius $ r $ is $ 3r^2 $.