SOLUTION: PQRS is a rectangle inscribed in a circle. The circle has centre O and radius r. The angle POQ is 120 degrees. Find the ratio of the circumference of the circle to the perimeter of

Question 1150434: PQRS is a rectangle inscribed in a circle. The circle has centre O and radius r. The angle POQ is 120 degrees. Find the ratio of the circumference of the circle to the perimeter of the rectangle.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Draw the figure as described, showing diagonals PR and QS intersecting at O. Add the two line segments through O parallel to the sides of the rectangle.

The rectangle is now divided into 8 congruent triangles. With the given measure of angle POQ, all of those triangles are 30-60-90 right triangles.

If the radius of the circle is r, then the lengths of the legs of each of those triangles are r/2 and (r*srqt(3))/2. That means the lengths of the sides of the rectangle are r and r*sqrt(3).

So the perimeter of the rectangle is

and the circumference of the circle is

The ratio of the circumference of the circle to the perimeter of the rectangle is then

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