Incircle — Definition, Formula & Examples

The incircle of a triangle is the circle that sits inside the triangle and is tangent to all three of its sides. Its center, called the incenter, is the point where the three angle bisectors of the triangle meet.

Given a triangle $\triangle ABC$ , the incircle is the unique circle inscribed within the triangle that is tangent to each side. Its center is the incenter $I$ , located at the intersection of the interior angle bisectors, and its radius $r$ (the inradius) satisfies $r = \dfrac{A}{s}$ , where $A$ is the area of the triangle and $s$ is its semi-perimeter.

Key Formula

r = \frac{A}{s} \quad \text{where} \quad s = \frac{a + b + c}{2}

Where:

$r$ = Inradius — the radius of the incircle
$A$ = Area of the triangle
$s$ = Semi-perimeter of the triangle
$a, b, c$ = Lengths of the three sides of the triangle

How It Works

To find the incircle, you need two quantities: the triangle's area and its semi-perimeter. The semi-perimeter is half the sum of the three side lengths. Dividing the area by the semi-perimeter gives the inradius. The incenter itself can be found by constructing the angle bisectors from any two vertices; their intersection is the center of the incircle.

Worked Example

Problem: Find the inradius of a triangle with sides 6, 8, and 10.

Identify the triangle: Since 6² + 8² = 36 + 64 = 100 = 10², this is a right triangle.

Compute the semi-perimeter: Add the three sides and divide by 2.

s = \frac{6 + 8 + 10}{2} = 12

Compute the area: For a right triangle, the legs serve as base and height.

A = \frac{1}{2}(6)(8) = 24

Apply the inradius formula: Divide the area by the semi-perimeter.

r = \frac{24}{12} = 2

Answer: The inradius is

r = 2

Why It Matters

The incircle appears frequently in competition geometry and in problems involving tangent lengths within triangles. Engineers and designers also use inscribed circles when determining the largest circular object that can fit inside a triangular region.

Common Mistakes

Mistake: Confusing the incircle (inscribed inside the triangle) with the circumscribed circle (passing through all three vertices).

Correction: The incircle is tangent to the sides and lies entirely within the triangle. The circumscribed circle (circumcircle) passes through the vertices and contains the triangle. Their radii use different formulas.