What is the area of the circle inscribed in an equilateral triangle of side $24$ cm?
- $24\ \pi\ cm^{2}$
- $36\ \pi\ cm^{2}$
- $48\ \pi\ cm^{2}$
- $18\ \pi\ cm^{2}$
Reveal answer
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C
Correct answer
Explanation
For an equilateral triangle with side 'a', the inradius 'r' is a / (2 * sqrt(3)). Here a = 24, so r = 24 / (2 * sqrt(3)) = 12 / sqrt(3) = 4 * sqrt(3). The area of the circle is pi * r^2 = pi * (4 * sqrt(3))^2 = pi * 16 * 3 = 48 * pi.
AI explanation
The formula for the radius of a circle inscribed in an equilateral triangle is r equals a divided by the quantity 2 times root 3, where a is the side length. Substituting the side length of 24 cm gives r equals 24 divided by 2 times root 3, which simplifies to 4 times root 3 cm. The area of the circle is pi times r squared, so we calculate pi times the quantity 4 times root 3 squared to get 48 pi square centimeters.