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An equilateral triangle is circumscribed inside a circle with a radius of 6. What is the area of the triangle?

Answer 1

Sadie Adams

Area of the equilateral triangle #color(green)(46.77#

Given triangle is equilateral and the radius of the circum circle is 6.

#a / sin A = b / sin B = c / sin C = 2R# where R is the radius of the circum circle.

#hat A = hat B = hat C = pi/3, a = b = c# being an equilateral triangle.

#:. a = b = c = 2R sin A = 2 * 6 * sin (pi/3) = 6sqrt3#

Area of triangle #A_t = (sqrt3/4) a^2 #

#A_t = (sqrt3/4) (6sqrt3)^2 = 27sqrt3 = color(green)(46.77#

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Answer 2

Noah Atkinson

The area of the equilateral triangle circumscribed inside a circle with a radius ( r ) can be calculated using the formula:

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

Given that the radius of the circle is 6, the side length of the equilateral triangle is twice the radius, so ( \text{side length} = 2 \times 6 = 12 ).

Substituting this into the formula:

[ \text{Area} = \frac{\sqrt{3}}{4} \times (12)^2 ]

[ \text{Area} = \frac{\sqrt{3}}{4} \times 144 ]

[ \text{Area} = 36\sqrt{3} ]

So, the area of the equilateral triangle is ( 36\sqrt{3} ) square units.

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Answer from HIX Tutor

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