SOLUTION: A square is inscribed in a circle, which in turn is inscribed in an equilateral triangle. If the area of the square is 16, what is the area of the region inside the triangle but ou

Algebra ->  Triangles -> SOLUTION: A square is inscribed in a circle, which in turn is inscribed in an equilateral triangle. If the area of the square is 16, what is the area of the region inside the triangle but ou      Log On


   

Question 1190330: A square is inscribed in a circle, which in turn is inscribed in an equilateral triangle. If the area of the square is 16, what is the area of the region inside the triangle but outside the circle?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


A diagram....



The medians (blue) of the equilateral triangle (black) meet at the centroid of the triangle, which is the center of both the circle (red) and the square (green). That point of intersection of the medians divides each median into two parts whose lengths are in the ratio 2:1. Then

(1) The side length of the square is given as 4, so the diagonal of the square is 4%2Asqrt%282%29.
(2) That makes the radius of the circle 2%2Asqrt%282%29.
(3) The diagonal of the square is 2/3 the length of the median of the triangle, so the median of the triangle (i.e., its height) is 6%2Asqrt%282%29.
(4) Use one of the 30-60-90 right triangles that is half of the triangle, along with the height of the triangle from (3), to determine that the side length of the triangle is %282%2Fsqrt%283%29%29%286%2Asqrt%282%29%29=4%2Asqrt%286%29.

Use basic area formulas to find the areas of the equilateral triangle with side length 4%2Asqrt%286%29 and the circle with radius 2%2Asqrt%282%29 and thus determine the area inside the triangle and outside the circle.