# Inscribed Circle of a Triangle Calculator

Calculate the properties of the inscribed circle within a triangle effortlessly using our calculator.

 Side a: Side b: Side c: Inscribed Circle Radius : Inscribed Circle Area : Inscribed Circle Perimeter :

## Similar Calculators:

An inscribed circle of a triangle, often referred to as the incircle, is a circle that is enclosed within a triangle, such that it touches all three sides of the triangle. This circle is unique because its center, called the incenter, is equidistant from all three sides of the triangle. The radius of the inscribed circle is known as the inradius.

1. Radius (r) = √s(s-a)(s-b)(s-c) / s
2. s = (a + b + c) / 2

## Properties of the Inscribed Circle

Let’s delve into the key properties of the inscribed circle of a triangle:

### 1. Tangency

The most defining property of the inscribed circle is that it touches all three sides of the triangle at a single point. This point of tangency is known as the point of tangency or the contact point. The inradius, which is the radius of the inscribed circle, is the distance from the incenter to the point of tangency.

In an inscribed circle, all three line segments drawn from the incenter to the points of tangency are of equal length. This means that the inradius is equidistant from all three sides of the triangle.

### 3. Relationship with Triangle’s Area

The area of the triangle can be calculated using the inradius and the semi-perimeter (half the sum of the triangle’s three sides). This relationship is given by the formula:

### 4. Incenter’s Location

The incenter of the triangle is the point of concurrency of the angle bisectors of the triangle. This means that the three angle bisectors intersect at the center of the inscribed circle.

### 5. Theorems and Applications

The inscribed circle is a fundamental concept in geometry and trigonometry. It is used in various theorems and proofs, including those related to the properties of triangles, trigonometric identities, and geometric constructions.