# Area of Square inscribed in circle

Ratio of Areas

A circle is inscribed in an equilateral triangle and the square is inscribed in the circle. Find the ratio of the area of triangle to the area of square

A diagram of the geometric figure used in the problem statement is shown below:

### Comments for Area of Square inscribed in circle

 Apr 16, 2013 Ratio of Areas by: Staff Answer Part I Calculate the radius of the inscribed circle: All three interior angles of the equilateral triangle = 60° Draw a straight line from the vertex of each interior angle so that each straight line bisects the angle. The three straight lines will cross at the exact center of the triangle, circle, and square. ------------------------------------------------

 Apr 16, 2013 Ratio of Areas by: Staff ------------------------------------------------ Part II The angle bisectors form six right triangles within the equilateral triangle. Each right triangle is a 30°-60°-90° triangle. Since the triangle is a 30°-60°-90° triangle, the lengths of each side of the triangle are known. If the radius of the circle is r (the short side of the triangle), the hypotenuse is 2r (twice the short side), and the length of the remaining side of the triangle is r√3. Since half of one of the diagonals of the square = r, the length of one of the diagonals = 2r. since 1/2 of a side of the equilateral triangle is r√3, the length of each side is 2r√3. ------------------------------------------------

 Apr 16, 2013 Ratio of Areas by: Staff ------------------------------------------------ Part III We now have all the information needed to compute the ratio asked for in the problem statement The area of the square can be calculated when the length of a diagonal is known. The area of the equilateral triangle can be calculated when the length of a side is known. ------------------------------------------------

 Apr 16, 2013 Ratio of Areas by: Staff ------------------------------------------------ Part IV The ratio of the Area of the equilateral triangle to the Area of the square is: Thanks for writing. Staff www.solving-math-problems.com