Question

If we inscribe a circle such that it is touching all six corners of a regular hexagon of side 10 inches, what is the area of the Circle that is not covered by the hexagon? ( with explanation )

Answer 1

`\left(100\pi - 150√(3)\right)` square inches.

Explanation:

Area of the Inscribed Hexagon

Refer to the first diagram attached. This inscribed regular hexagon can be split into six equilateral triangles. The length of each side of these triangle will be
`10` inches (same as the length of each side of the regular hexagon.)

Refer to the second attachment for one of these equilateral triangles.

Let segment
`\sf CH` be a height on side
`\sf AB`. Since this triangle is equilateral, the size of each internal angle will be
`\sf 60^\circ`. The length of segment

`\displaystyle 10\, \sin\left(60^\circ\right) = 10 * (√(3))/(2) = 5√(3)`.

The area (in square inches) of this equilateral triangle will be:

`\begin{aligned}&(1)/(2) * \text{Base} *\text{Height} \\ &= (1)/(2) * 10 * 5√(3)= 25√(3) \end{aligned}`.

Note that the inscribed hexagon in this question is made up of six equilateral triangles like this one. Therefore, the area (in square inches) of this hexagon will be:

`\displaystyle 6 * 25√(3) = 150√(3)`.

Area of of the circle that is not covered

Refer to the first diagram. The length of each side of these equilateral triangles is the same as the radius of the circle. Since the length of one such side is
`10` inches, the radius of this circle will also be
`10` inches.

The area (in square inches) of a circle of radius
`10` inches is:

`\pi * (\text{radius})^2 = \pi * 10^2 = 100\pi`.

The area (in square inches) of the circle that the hexagon did not cover would be:

`\begin{aligned}&\text{Area of circle} - \text{Area of hexagon} \\ &= 100\pi - 150√(3)\end{aligned}`.