SOLUTION: A circle is inscribed inside an equilateral triangle. If the circumference of the circle is 6 cm, then the area of the triangle, in cm^2 is:
A) 27 √3 π
B) 27 √3/π
C) 27
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Question 1167551: A circle is inscribed inside an equilateral triangle. If the circumference of the circle is 6 cm, then the area of the triangle, in cm^2 is:
A) 27 √3 π
B) 27 √3/π
C) 27 √3/π^2
D) 9 √3/π^2
E) 9 √3 π
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
Draw the figure.
Draw the three altitudes of the equilateral triangle, intersecting at the center of the circle. Those altitudes divide the equilateral triangle into six congruent 30-60-90 right triangles, in which the ratio of the side lengths is 1:sqrt(3):2.
(1) The circumference of the circle is 6; find the radius using .
(2) The radius of the circle is the short leg of one of the 30-60-90 right triangles. Use the radius from (1) and the ratio 1:sqrt(3):2 to find the long leg of each of the 30-60-90 triangles.
(3) The side length of the triangle is twice the length found in (2).
(4) The area of an equilateral triangle with side length s is .
If you do all those calculations correctly, you should finish with answer choice C.
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