Question

A circle inscribed in an equilateral triangle. If the circumference of circle is 176 cm, find the area of the shaded region.

Answer 1

check out the screen-shot may it helpful...

Answer 2

Let the Side of the Given Equilateral Triangle be : S

We know that :

(1). If 'S' is the side of an Equilateral Triangle then the Altitude of the Equilateral Triangle will be √3/2 × S

(2). In an Equilateral Triangle: Altitudes are same as Medians and pass through the center of Circle inscribed in it, because the Centroid of an Equilateral Triangle is same as the Center of the Circle inscribed in it.

(3). A Centroid divides the Median in the Ratio of 2 : 1 and as the Centroid is same as Center of Circle in Equilateral Triangle, We can say that the Radius of the inscribed Circle will be 1/3 rd of the Altitude(Median)

Let us take an Altitude from Vertex A to the Side BC which is of length √3/2 × S

As the Centroid(Centre of Circle) Divides this Altitude in the ratio of 2 : 1

The Length of one part (of the three parts) of Altitude is √3/2 × S × 1/3 = √3/6 × S

⇒ The Radius of Inscribed Circle (r) = √3/6 × S

We know that Circumference of a Circle = 2πr

Given the Circumference of the Circle = 176 cm

⇒ 2πr = 176

⇒ 22/7 × r = 88

⇒ r = 28cm

But we know that r = √3/6 × S

⇒ √3/6 × S = 28

⇒ S = 168/√3

We Know that the Area of an Equilateral Triangle = √3/4 × S²

⇒ The Area of Given Triangle = √3/4 × (168)²/3 = 2352√3 = 4073.664 cm²

We know that Area of a Circle = πr² = 22/7 × 28 × 28 = 2464 cm²

⇒ The Area of the Shaded Region = Area of the Given Triangle - Area of the Circle

⇒ The Area of the Shaded Region = 4073.664 - 2464 = 1609.664 cm²