Inscribed Circle in a Triangle — Definition, Formula & Examples

The inscribed circle (or incircle) of a triangle is the largest circle that fits entirely inside the triangle, touching all three sides exactly once. Its center, called the incenter, is the point where the triangle's three angle bisectors meet.

Given a triangle $\triangle ABC$ , the inscribed circle is the unique circle tangent to all three sides of the triangle. Its center $I$ is the point of concurrency of the three interior angle bisectors, and its radius $r$ is the perpendicular distance from $I$ to any side.

Key Formula

r = \frac{A}{s}

Where:

$r$ = Inradius — the radius of the inscribed circle
$A$ = Area of the triangle
$s$ = Semi-perimeter of the triangle: $s = \frac{a + b + c}{2}$, where $a$, $b$, $c$ are the side lengths

How It Works

To construct the incircle, bisect any two interior angles of the triangle; their intersection is the incenter

I

. Drop a perpendicular from

I

to any side to find the inradius

r

. Draw a circle centered at

I

with radius

r

, and it will be tangent to all three sides. You can also calculate

r

directly from the triangle's area and perimeter using the formula below.

Worked Example

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

Step 1: Compute the semi-perimeter.

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

Step 2: Find the area. Since 6-8-10 is a right triangle (check:

6^2 + 8^2 = 100 = 10^2

), the area is half the product of the legs.

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

Step 3: Apply the inradius formula.

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

Answer: The inscribed circle has a radius of 2 units.

Why It Matters

The incircle appears in competition geometry problems and standardized tests whenever you need to relate a triangle's area to its side lengths. Engineers and architects use inscribed circles when fitting circular objects inside triangular frames or supports.

Common Mistakes

Mistake: Confusing the incenter (intersection of angle bisectors) with the circumcenter (intersection of perpendicular bisectors).

Correction: The incenter is inside every triangle and is the center of the inscribed circle. The circumcenter is the center of the circumscribed circle and can lie outside the triangle for obtuse triangles.