Question

1. A hexagon is inscribed in a circle. If the difference between the area of the circle and the area

of the hexagon is 24 m”, use the formula for the area of a sector to approximate the radius
of the circle. Give an exact answer( doesn't have to be rationalized) and round to 2 decimals.

Answer 1

The radius, r ≈ 6.55 m

Explanation:

The given parameters are;

An inscribed hexagon in a circle

The area of the hexagon = Area of the circle - 24 m²

The formula for the area of the circle = π·r² = π × r²

The area of each of the six equilateral triangle in the hexagon = √3/4 × r²

Where;

r = The radius of the circle

The area of a sector of a circle in which the equilateral triangle is inscribed = 60/360×π·r² = 1/6×π·r² = 1/6 × Area of the circle

Given that the sector of the circle in which one of the six equilateral triangles of the hexagon is inscribed is 1/6 the area of the circle, the difference in area between the equilateral triangle and the sector = 24/6 = 4

Therefore, the difference in the area between the sector of the circle and the area of the equilateral triangle = 4 m²

We then write; 1/6×π·r²- √3/4 × r² = 4

r²·(1/6×π- √3/4) = 4

r² = 4/(1/6 ×π - √3/4) = 48/(2·π -3·√3)

r = √(48/(2·π -3·√3) = 6.55 m to 2 decimal places

The radius, r ≈ 6.55 m.