Inscribed Circle of a Triangle Calculator

What is an Inscribed Circle?

An inscribed circle of a triangle is a circle that is tangent to each side of the triangle. The center of the inscribed circle is called the incenter, and the radius of the inscribed circle is called the inradius.

Formulas for Inscribed Circle

Let \(a\), \(b\), and \(c\) be the side lengths of the triangle, \(s\) be the semi-perimeter, \(A\) be the area, and \(r\) be the radius of the inscribed circle. Then:

1. Semi-perimeter: \(s = \frac{a + b + c}{2}\)
2. Area: \(A = \sqrt{s(s - a)(s - b)(s - c)}\)
3. Radius: \(r = \frac{A}{s}\)

Step-by-Step Calculations

Let's calculate the radius of the inscribed circle for a triangle with side lengths \(a = 5\), \(b = 6\), and \(c = 7\) units:

1. Semi-perimeter: \[s = \frac{5 + 6 + 7}{2} = 9 \text{ units}\]
2. Area: \[A = \sqrt{9(9 - 5)(9 - 6)(9 - 7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} = 14.7 \text{ square units}\]
3. Radius: \[r = \frac{14.7}{9} = 1.63 \text{ units}\]

This diagram illustrates the triangle with the inscribed circle and its radius.