A circle is inscribed in an isosceles triangle. Find its radius if the triangle’s height is 18.

An equilateral (or regular) triangle is a triangle in which all sides and angles are equal (angles of 60 °).
1. The length of the height of a regular triangle is found by the formula:
h = (a√3) / 2,
where a is the length of the side of the triangle.
Substitute the data according to the value condition and find the length of the side of a regular triangle:
(a√3) / 2 = 18;
a = (2 * 18) / √3 (proportional);
a = 36 / √3.
Let’s get rid of irrationality in the denominator:
a = 36 / √3 * √3 / √3 = (36 * √3) / (√3 * √3) = (36√3) / (√3) ² = 36√3 / 3 = 12√3.
2. The length of the radius of a circle inscribed in a regular triangle is found by the formula:
r = (a√3) / 6.
Substitute the known values ​​and find the length of the inscribed circle radius:
r = (12√3 * √3) / 6 = (12 * (√3) ²) / 6 = (12 * 3) / 6 = 36/6 = 6.
Answer: r = 6.