The area of the inscribed equilateral triangle is 75/4*sqrt(3). Find the perimeter of circumscribing square to the nearest whole number.
I'm assuming that the area of the triangle is (75/4) * sqrt (3)
We can solve for the side length, S, of the triangle as follows
(75/ 4)sqrt (3) = (1/2) S^2 sin (60°)
(75/4)sqrt (3) = (1/2) S^2 * sqrt (3) / 2
(75/4) = (1/4)S^2
75 = S^2
sqrt (75) = S
And using the Law of Cosines we can find the radius, R, of the circle
75 = 2R^2 - 2R^2 cos (120°)
75 = 2R^2 + R^2
75 = 3 R^2
25 = R^2
5 = R
And the side of the square is twice this = 10
So.....the perimeter of the square = 4 * 10 = 40 units
The area of the inscribed equilateral triangle is 75/4*sqrt(3). Find the perimeter of circumscribing square to the nearest whole number.
The ratio of the area of an equilateral triangle to its circumcircle's area is: 1.299 : pi or 0.413496
Area of the triangle is 75/4*sqrt(3) = 32.47595 u²
Area of the circle is 32.47595 / 0.413496 = 78.54 u²
Raduis of the circle is r = sqrt( 78.54 / pi ) = 5
Perimeter of a square is P = 8r = 40 units