Question

What radius of a circle is required to inscribe a regular hexagon with an area of 64.95 cm2 and a apothem of 4.33 cm

Answer 1

The radius is 5 cm.

Explanation:

Given the area of regular hexagon which is inscribed in the circle. we have to find the radius of circle which is required to inscribe a regular hexagon with an area of 64.95 square centimeter and a apothem of 4.33 cm

The apothem of a regular hexagon is a line from the center to the midpoint of one of side of hexagon.

Also, central angle of regular hexagon is of 60° which gives the angle ∠ACO=30°

Given, OC=4.33 cm

In ΔAOC

`cos30^(\circ)=(OC)/(OB)=(4.33)/(OB)`

⇒
`(\sqrt3)/(2)=(4.33)/(OB)`

⇒
`OB=(4.33)/(\sqrt3)* 2=5 cm`

To verify, we find the area of hexagon with the given side=radius=5 cm

`\text{Area of hexagon=}(3\sqrt3)/(a^2)=(3\sqrt3)/(25)=64.95cm^2`

Answer 2

Area of the hexagon is given as 64.95cm^2 and the apothem is 4.33m. The apothem divides a side of the hexagon into two equal parts. Drawing a triangle this side fro the center of the hexagon where the central angle would be 60 degrees which would lead into the conclusion that the triangle is equilateral and since the apothem divides this further into two we will have a right triangle. We use pythagorean theorem to solve the unknown side as follows:

(2x)^2 = x^2 + 4.33^2

wherre x is one half the side of the hexagon, 2x would be the radius.

x = 2.50

Therefore, the radius would be 2.50x2 = 5 cm