An equilateral triangle is given, the side of which is 15 cm. A circle is inscribed in the triangle.

An equilateral triangle is given, the side of which is 15 cm. A circle is inscribed in the triangle. Calculate the area of the inscribed circle.

1. It is known that the sum of the angles of a triangle is 180 degrees.

It is known that the center of a circle inscribed in a triangle lies at the intersection point of the bisectors of the triangle.

It is known that in an equilateral triangle, each bisector is both the height and the median.

2. In an equilateral triangle ABC from the vertex B we draw the bisector BM to the side AC, and from the vertex A we draw the bisector AN. Got the point of their intersection point O.

In the formed triangle AOM, the angle of AMO is straight (since BM is the height).

3. Determine the angle OAM in the triangle AOM.

Angle A = 180: 3 = 60 degrees.

OAM angle = 60: 2 = 30 degrees.

4. In the triangle AOM, find the side AM.

AM = CA: 2 = 15 cm: 2 = 7.5 cm.

5. Find the leg OM of the AOM triangle, which lies opposite an angle of 30 degrees by the tangent of the angle.

tg ОАМ = 3 ^ 1/2/3.

ОМ / АМ = tan ОАМ = 3 ^ 1/2: 3;

OM = 3 ^ 1/2: 3 * AM = 3 ^ 1/2: 3 * 7.5 = 3 ^ 1/2 * 2.5.

6. The area of ​​the inscribed circle is equal to P * R ^ 2 = P * OM ^ 2 =

3 * (3 ^ 1/2 * 2.5) ^ 2 = 3 * 3 * 6.25 = 56.25 cm ^ 2.

Answer: The area of ​​the inscribed circle is 56.2 5 cm ^ 2.