Can you find the area of the shaded region?

Geometry Level 2

A circle is inscribed in an equilateral triangle of side length 6 6 as shown. Find the area of the shaded region correct to two decimal places. Use π = 22 7 \pi=\dfrac{22}{7} .


The answer is 6.16.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

tan ⁡ 30 = r 3 \tan 30 = \dfrac{r}{3} ⟹ \implies r = 3 tan ⁡ 30 r=3~\tan 30

Area of the shaded region is equal to the area of the equilateral triangle minus the area of the circle.

A = 3 4 ( 6 2 ) − 22 7 ( 3 tan ⁡ 30 ) 2 ≈ 6.16 square units A=\dfrac{\sqrt{3}}{4}(6^2) - \dfrac{22}{7}(3~\tan 30)^2 \approx \boxed{6.16~\text{square units}}

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Could we, please, use π \pi as π \pi instead of some other number being preferred for it. Thank you.

Marta Reece - 3 years, 5 months ago

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We have different calculators, that's why I put an approximation for generality purposes.

A Former Brilliant Member - 3 years, 5 months ago

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