A regular decagon(12 sides) in inscribed in a circle with the radius r. The decagon has an area of108 in^2. What is the radius of the cirlce?
To find the radius of the circle inscribed with a regular decagon, we can use the following formula:
Radius of the inscribed circle = (Side length of the decagon) / (2 * tan(π/12))
First, we need to find the side length of the decagon. To do this, we can use the formula:
Area of the decagon = (5/2) * (Side length^2) * tan(π/12)
Given that the area of the decagon is 108 in^2, we can rearrange the formula to solve for the side length:
108 = (5/2) * (Side length^2) * tan(π/12)
Dividing both sides of the equation by (5/2) * tan(π/12), we get:
Side length^2 = 108 / [(5/2) * tan(π/12)]
Now, we can calculate the side length:
Side length = √[108 / (5/2) * tan(π/12)]
Once we have the value of the side length, we can substitute it into the formula for the radius of the inscribed circle:
Radius of the inscribed circle = (Side length) / (2 * tan(π/12))
Calculating this expression will give us the radius of the circle.