A circle has a radius of 6 in. Find the area of its inscribed equilateral triangles.

Question

anonymous · Answer

The circumscribed equilateral triangle will have an area of:

hero · Answer

If a is one of the sides of the equilateral, then: 

$$r = a \frac{\sqrt{3}}{6}$$

If r = 6, simply substitute r into the equation and solve for a. Then use Heron's formula to find the area.

directrix · Answer

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Area of inscribed triangle = 1/2 ap where a is the apothem and p is the perimeter

A = 1/2 (3) (6 √ 3) (3) = 27 √ 3

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Area of Circumscribed Triangle
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A = 1/2 a (p)

A = 1/2 (6) (12√ 3) * 3

A = 108 √ 3

Note. The circumscribed and inscribed  equilateral triangles are similar. The apothem of the inscribed triangle is 3 and the apothem of the circumscribed triangle is 6. The ratio of the areas of the triangles is (3/6)^2 - (1/2)^2 = 1/4. Because the area of the inscribed triangle is 27√ 3, the area of the circumscribed triangle is 108 √ 3.

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