Question

A circle is inscribed in an equilateral triangle of side 2424 cm, touching its sides. What is the area of the remaining portion of the triangle?

Answer 1

Question : In an equilateral triangle of side 24 cm, a circle is inscribed touching it's sides. find the area of remaining portion of the triangle

Answer :

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

Answer 2

Answer::

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.

Step-by-step explanation::

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