To find the area of the circle inside the equilateral triangle, we can use the concept of the incircle of a triangle.

Let's consider an equilateral triangle with side length 21 cm. The triangle is divided into three congruent smaller triangles, each with side length 21 cm/3 = 7 cm.

Let's denote the radius of the circle as r.

Key Point 1: Radius of the circle bisects the side of the triangle at a right angle
Since the circle touches all three sides of the triangle, the radius of the circle bisects each side of the triangle at a right angle. Therefore, we can draw three right-angled triangles within the equilateral triangle.

Key Point 2: Height of the right-angled triangle is r√3/2
In an equilateral triangle, the altitude (height) is given by the formula h = a√3/2, where a is the side length of the triangle. In this case, the side length is 7 cm. Therefore, the height of each right-angled triangle is r√3/2.

Key Point 3: Hypotenuse of the right-angled triangle is r
The hypotenuse of the right-angled triangle is the radius of the circle, which we denoted as r.

Key Point 4: Area of the right-angled triangle is (1/2) * base * height
The area of each right-angled triangle can be calculated using the formula for the area of a right-angled triangle: (1/2) * base * height. In this case, the base is r and the height is r√3/2.

Key Point 5: Total area of the equilateral triangle is 3 times the area of one right-angled triangle
Since the equilateral triangle is made up of three congruent right-angled triangles, the total area of the equilateral triangle is 3 times the area of one right-angled triangle.

Key Point 6: Area of the equilateral triangle is (21^2 * √3)/4
The area of an equilateral triangle can be calculated using the formula: (√3/4) * a^2, where a is the side length of the triangle. In this case, the side length is 21 cm.

Key Point 7: Area of the circle is (1/3) * area of the equilateral triangle
Since the area of the equilateral triangle is 3 times the area of one right-angled triangle, the area of the circle is (1/3) times the area of the equilateral triangle.

Now, let's put all these key points together to find the area of the circle.

Solution:
- The area of the equilateral triangle = (√3/4) * a^2 = (√3/4) * 21^2 = (√3/4) * 441 = (441√3)/4
- The area of one right-angled triangle = (1/2) * base * height = (1/2) * r * (r√3/2) = (r^2√3)/4
- The total area of the equilateral triangle = 3 times the area of one right-angled triangle = (3 * (r^2√3)/4) = (3r^2