Question

A hexagon is inscribed in a circle. If the difference between the area of the circle and the area of the hexagon is 36 m2, use the formula for the area of a sector to approximate the radius r of the circle. (Round your answer to three decimal places.)

Answer 1

Using just the equation for the area of a sector doesn't really help. Here's why... the area of a sector is:

A = (1/2)r²θ

where r is the radius and θ is the central angle (in radians). If you want to do it in degrees, then you have to remember that π radians = 180 degrees, so the area formula for θ in degrees would be:

A = (1/2)r² [θ * π/180]

A hexagon has 6 sides, which means that you can break the circle into 6 sectors. Each sector will have 60 degrees (π/3 radians). So, the area of ONE of these sectors will be:

A = (1/2)r²(π/3)

And there will be SIX sectors, so the total area of the circle will (obviously) be:

A = 6 * (1/2)r²(π/3) = 3r²(π/3) = πr²

This isn't very helpful, so I think you are supposed to use the formula for a SEGMENT of a circle. You can find this by taking the formula for the area of the sector and subtracting out the area of the isosceles triangle. This will give you the area that is inside the circle, but outside the hexagon.

To the area of each of the six triangle sections of the hexagon is easy to find in terms of the radius, because when you draw in the "height" of each, you will form a 30-60-90 right triangle with the "height" as the long leg, half the "base" as the short leg, and the radius as the hypotenuse. Remembering that the ratio of the sides of a 30-60-90 triangle is:

1:√3:2 ---> (1/2)base:height:radius

So...

height = (√3/2) radius

(1/2) base = (1/2) radius

So, the area of the triangle will be:

A = (1/2) base * height

A = (1/2) r * (√3/2) r = (1/2)(√3/2)r²

You'll see why I left the (1/2) separate in the next step. The formula for area of the SEGMENT is then, the total area of the SECTOR minus the area of the triangle:

A = (1/2)r²(π/3) - (1/2)(√3/2)r²

A = (1/2)r² [(π/3) - (√3/2)]

So we know that the total area outside the hexagon will be 30 m², and there are going to be 6 segments to add up, and we get:

30 = 6 * (1/2)r² [(π/3) - (√3/2)]

30 = 3 * r²[(π/3) - (√3/2)]

10 = r²[(π/3) - (√3/2)]

r = √ (10 / [(π/3)-(√3/2)] ) ≈ 7.429 meters

Hope that helps

Answer 2

total area =36=6*(area of any iner triangle lets say A1)
A1=6
\[A_1=\frac{ r^2\sqrt{3} }{ 4 }=6\] such that r is the radius