GeomBench

Inradius Calculator

The largest circle that fits inside your shape.

Side a

Side b

Side c

Inradius

1.633

What you entered

Sides: 5, 6, 7

1
Semiperimeter s = (a + b + c) ÷ 2
(5 + 6 + 7) ÷ 2= 9
2
Area (Heron's formula) = √(s(s−a)(s−b)(s−c))
√(9 × 4 × 3 × 2)= 14.6969
3
Inradius = Area ÷ s
14.6969 ÷ 9= 1.633

Result

Inradius: 1.633

The inradius (radius of the largest circle that fits inside) is 1.633 units.

Inradius of triangles and regular polygons

The inradius is the radius of the largest circle fitting inside the shape, touching every side. Circumradius is shown for comparison.

Shape	Dimensions	Inradius	Circumradius
Triangle	3, 4, 5	1.0000	2.5000
Triangle	6, 8, 10	2.0000	5.0000
Triangle	5, 12, 13	2.0000	6.5000
Polygon	6 sides, side 4	3.4641	4.0000
Polygon	5 sides, side 6	4.1291	5.1039

For a right triangle the circumradius is always exactly half the hypotenuse, which the first three rows confirm - 2.5 for a hypotenuse of 5, 5 for 10, 6.5 for 13. That follows from Thales' theorem: the hypotenuse is a diameter of the circumscribed circle. The inradius comes from area divided by the semi-perimeter. For a regular polygon the inradius is the apothem, the perpendicular distance from centre to side, and as the number of sides grows both radii converge on each other and the shape approaches a circle.

Area divided by half the perimeter

A triangle's inradius is its area divided by its semiperimeter (r = Area ÷ s) — a direct consequence of the fact that the incircle touches all three sides, splitting the triangle into three smaller triangles whose combined area equals the whole.

The regular-polygon shortcut

For a regular polygon, the inradius is also called the apothem, and it has a clean closed form (s ÷ (2·tan(π/n))) — no need for the more general triangle formula.

Frequently asked questions

What is the inradius of a triangle?

The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides. Formula: r = Area / s, where s = (a+b+c)/2 is the semi-perimeter. For a 3-4-5 triangle: Area = 6, s = 6, r = 1.

What is an apothem?

The apothem of a regular polygon is the perpendicular distance from its center to any side — it's the same as the inradius. For a regular hexagon with side 10: apothem = 10 / (2×tan(π/6)) = 10 / (2×0.577) = 8.66.

How is inradius used in geometry?

The incircle helps determine the largest round pipe or rod that fits inside a triangular cross-section, calculate inscribed patterns in polygonal designs, and appears in optimization problems where you need the largest circle inside a given shape.