The Area of a Circle Inscribed in an Equilateral Triangle

To find the perimeter of an equilateral triangle, we need to determine the length of one side. In this problem, we are given the area of a circle inscribed in an equilateral triangle and we will use this information to find the perimeter of the triangle.

Given:
- Area of the circle inscribed in the equilateral triangle = 154 cm²
- π (pi) = 22/7
- √3 (root 3) = 1.73

Step 1: Finding the Radius of the Circle
The area of a circle can be calculated using the formula A = πr², where A is the area and r is the radius. In this case, we are given the area of the circle, so we can rearrange the formula to solve for the radius.

A = πr²
154 = (22/7)r²
154 * 7/22 = r²
49 = r²

Taking the square root of both sides, we find:
r = √49
r = 7 cm

Therefore, the radius of the circle is 7 cm.

Step 2: Finding the Length of the Side of the Equilateral Triangle
In an equilateral triangle, the radius of the inscribed circle is equal to one-third of the altitude of the triangle. Since the altitude of an equilateral triangle bisects the base, we can use this information to find the length of the side of the triangle.

Let x be the length of the side of the equilateral triangle. The altitude will then be √3/2 * x.

Since the radius of the circle is equal to one-third of the altitude, we have:
7 = (1/3)(√3/2 * x)
7 = √3/6 * x

Multiplying both sides by 6/√3, we get:
7 * 6/√3 = x
42/√3 = x
14√3 = x

Therefore, the length of one side of the equilateral triangle is 14√3 cm or approximately 24.25 cm.

Step 3: Finding the Perimeter of the Equilateral Triangle
The perimeter of an equilateral triangle is simply the sum of the lengths of its three sides. Since all sides of an equilateral triangle are equal, the perimeter is 3 times the length of one side. Using the value we found in Step 2:

Perimeter = 3 * 14√3
Perimeter = 42√3 cm or approximately 72.75 cm

Therefore, the perimeter of the equilateral triangle is 42√3 cm or approximately 72.75 cm.