A circle is inscribed in an equilateral triangle of side
24 centimetre, touching its sides. What is the area of the remaining portion of the triangle in square centimetre?
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refer to the attachment
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Step-by-step explanation:
Let r is radius of inscribed circle touching sides of given equilateral triangle.
Let ABC is given equilateral triangle, and O is the incentre of triangle.
then, area of equilateral triangle ABC = area of ∆AOB + area of ∆BOC + area of ∆COA.
or, √3/4 × (side)² = 1/2 × r × AB + 1/2 × r × BC + 1/2 × r × CA
or, √3/4 × (24)² = 1/2 × r(AB + BC + CA)
or, √3/4 × 24 × 24 = 1/2 × r (24 + 24 + 24)
or, 144√3 = 36r
or, r = 4√3 = 4√3 cm
so, area of inscribed circle = πr²
= π × (4√3)² = 48π cm²
area of equilateral triangle = √3/4 × 24²
= 144√3 cm²
area of remaining portion of the triangle = 144√3 cm² - 48π cm²
= 98.69 cm²
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