Inradius — Definition, Formula & Examples

Inradius

The radius of a plane figure's inscribed circle. For a regular polygon, this is the same as the apothem.

Incscribed Circle of a Triangle
Triangle with incenter marked at intersection of three angle bisectors, inscribed circle shown with three inradius segments...

See also

Incenter

Key Formula

r = \frac{A}{s}

Where:

$r$ = Inradius of the triangle
$A$ = Area of the triangle
$s$ = Semi-perimeter of the triangle, equal to (a + b + c) / 2

Worked Example

Problem: Find the inradius of a triangle with side lengths 3, 4, and 5.

Step 1: Compute the semi-perimeter.

s = \frac{3 + 4 + 5}{2} = 6

Step 2: Compute the area. Since this is a right triangle (3² + 4² = 5²), the area is half the product of the two legs.

A = \frac{1}{2}(3)(4) = 6

Step 3: Apply the inradius formula.

r = \frac{A}{s} = \frac{6}{6} = 1

Answer: The inradius of the 3-4-5 right triangle is 1.

Another Example

Problem: Find the inradius of a regular hexagon with side length 6.

Step 1: For a regular polygon with n sides of length a, the inradius (apothem) is given by the formula below.

r = \frac{a}{2\tan\!\left(\frac{\pi}{n}\right)}

Step 2: Substitute n = 6 and a = 6.

r = \frac{6}{2\tan\!\left(\frac{\pi}{6}\right)} = \frac{6}{2 \cdot \frac{1}{\sqrt{3}}} = \frac{6}{\frac{2}{\sqrt{3}}} = \frac{6\sqrt{3}}{2} = 3\sqrt{3}

Step 3: Approximate the result.

r = 3\sqrt{3} \approx 5.196

Answer: The inradius of a regular hexagon with side length 6 is 3√3 ≈ 5.196.

Frequently Asked Questions

How do you find the inradius of a triangle?

Use the formula r = A / s, where A is the area and s is the semi-perimeter (half the sum of all three sides). You can find the area using Heron's formula if you only know the side lengths: A = √[s(s − a)(s − b)(s − c)].

What is the difference between the inradius and the circumradius?

The inradius is the radius of the inscribed circle (the largest circle fitting inside the polygon, tangent to every side). The circumradius is the radius of the circumscribed circle (the smallest circle that passes through every vertex of the polygon). For any triangle, the circumradius is always greater than or equal to twice the inradius, with equality only for equilateral triangles.

Inradius vs. Circumradius

The inradius (r) measures from the incenter to the nearest side; the circumradius (R) measures from the circumcenter to any vertex. The inscribed circle sits inside the polygon; the circumscribed circle wraps around the outside. For a triangle, these are linked by the Euler inequality R ≥ 2r, with equality only when the triangle is equilateral.

Why It Matters

The inradius appears frequently in geometry competitions and in formulas connecting a triangle's area to its side lengths. It is also essential in engineering and design: the inscribed circle represents the largest circle you can cut from a polygonal piece of material. Understanding the inradius deepens your grasp of how a polygon's area, perimeter, and shape relate to one another.

Common Mistakes

Mistake: Confusing the inradius with the circumradius and using the wrong formula.

Correction: The inradius uses r = A / s (area over semi-perimeter). The circumradius uses R = abc / (4A). These are different quantities—the inradius is always smaller for any triangle.

Mistake: Forgetting to halve the perimeter when computing the semi-perimeter.

Correction: The semi-perimeter is s = (a + b + c) / 2, not a + b + c. Using the full perimeter in the formula will give you an inradius that is half the correct value.

Related Terms

Inscribed Circle — The circle whose radius is the inradius
Incenter — Center point of the inscribed circle
Radius — General concept that inradius is a specific case of
Apothem — Equals the inradius for regular polygons
Regular Polygon — Polygon where inradius equals the apothem
Plane Figure — Any flat shape that can have an inscribed circle
Circumscribed Circle — The outer circle, contrasted with the inscribed circle