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A circle has a radius of 6 inches. What would be the area of an inscribed equilateral triangle?

Answer 1

Ella Armstrong

Area = 27#sqrt(3)#

Length of a side of an equilateral triangle inscribed in a circle = #rsqrt(3)# , where r is the radius of the circle

Therefore, Area = #sqrt(3)a^2/4#
# a = 6sqrt(3)#

Note: how to get the above relation?

#a/sinA= c/sinC#

#=> c=a*(sinC/sinA) = 6*(sin 120/sin 30) = 6sqrt(3) #

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Answer 2

Mason Adams

To find the area of an equilateral triangle inscribed in a circle, we can use the formula:

[ \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 ]

For an equilateral triangle inscribed in a circle, the radius of the circle is equal to the distance from the center of the circle to any vertex of the triangle.

Given that the radius of the circle is 6 inches, each side of the equilateral triangle will also be 6 inches.

Substitute the side length into the formula:

[ \text{Area} = \frac{\sqrt{3}}{4} \times 6^2 ] [ = \frac{\sqrt{3}}{4} \times 36 ] [ = \frac{\sqrt{3} \times 36}{4} ] [ = \frac{36\sqrt{3}}{4} ] [ = 9\sqrt{3} ]

So, the area of the inscribed equilateral triangle is (9\sqrt{3}) square inches.

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Answer from HIX Tutor

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