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Inradius of Triangle

Geometry

Input Mode

Enter all three side lengths. Inradius is computed via Heron's formula: r = Area / s.

Side a

Side b

Side c

Length Unit

Decimal Places (0–10)

Inradius r

1.0000 cm

Right

Triangle

All Results

Semi-perimeter (s)6.0000 cm
Triangle Area
6.0000 cm²
Incircle Circumference (2πr)
6.2832 cm
Incircle Area (πr²)
3.1416 cm²
Triangle TypeRight

Incircle Diagram

rIABCIncircle (tangent to all three sides)

About This Tool

Inradius of Triangle – Inscribed Circle Radius Calculator

The inradius of a triangle is the radius of its inscribed circle (also called the incircle) — the largest circle that fits perfectly inside the triangle and is tangent to all three of its sides. This calculator finds the inradius instantly from a range of input types: three side lengths (SSS), two sides and an included angle (SAS), two angles and a side (AAS/ASA), or just the two legs of a right triangle.

What Is the Incircle and Incenter?

Every non-degenerate triangle has exactly one incircle. Its center is called the incenter (labeled I), which is the point where the three angle bisectors of the triangle meet. The distance from the incenter to each side of the triangle is constant and equal to the inradius r.

The incircle is an important concept in classical geometry. It appears in problems involving the largest circle that can be inscribed in a given triangular region — for example, placing a circular component inside a triangular frame, or computing tangent lengths from a vertex to the incircle.

The Core Formula: r = Area / s

The inradius is calculated using two quantities derived from the three side lengths a, b, and c:

  • Semi-perimeter: s = (a + b + c) / 2
  • Area (Heron's formula): A = √(s · (s−a) · (s−b) · (s−c))
  • Inradius: r = A / s

This formula comes from a neat geometric proof: draw lines from the incenter I to each of the three sides. These lines are perpendicular to their respective sides and each has length r. The triangle is thereby split into three smaller triangles, each with base equal to one side and height r. Summing the areas of these three triangles gives A = r · s, which rearranges to r = A / s.

Right Triangle Shortcut: r = (a + b − c) / 2

For a right triangle with legs a and b and hypotenuse c, the inradius formula simplifies dramatically. Because c² = a² + b² (Pythagorean theorem), substituting the area A = ab/2 and the semi-perimeter s = (a+b+c)/2 into r = A/s and simplifying algebraically yields:

r = (a + b − c) / 2

For the classic 3-4-5 right triangle, this gives r = (3 + 4 − 5) / 2 = 1. This shortcut is available in the Right Triangle mode of this calculator.

Alternate Input Modes

Not all triangles are given as three side lengths. This calculator supports four input modes:

  • SSS (Three Sides): All three sides a, b, c are known. The calculator validates the triangle inequality and applies Heron's formula directly.
  • SAS (Two Sides + Included Angle): Two sides and the angle between them are known. The missing third side is found using the Law of Cosines: c² = a² + b² − 2ab·cos(C), then the SSS path is followed.
  • AAS / ASA (Two Angles + One Side): Two angles and one side are known. The third angle is C = 180° − A − B, and the remaining sides are found via the Law of Sines: b = a·sin(B)/sin(A).
  • Right Triangle: The two legs are entered directly and the shortcut formula r = (a + b − c) / 2 is applied, where c = √(a² + b²).

Relationship Between Inradius and Circumradius

Every triangle also has a circumradius R — the radius of the circumscribed circle that passes through all three vertices. A classical result in geometry states that for any triangle, R ≥ 2r, with equality only when the triangle is equilateral. The two radii are related by Euler's formula:

OI² = R(R − 2r)

where O is the circumcenter and I is the incenter. This formula implies R ≥ 2r (since OI² ≥ 0), and it is used in advanced competition geometry to relate the positions of triangle centers.

Tangent Lengths from Vertices

A useful consequence of the incircle is that the tangent lines from each vertex to the incircle have equal lengths. If the incircle is tangent to side BC at point D, to CA at E, and to AB at F, then:

  • AF = AE = s − a
  • BF = BD = s − b
  • CD = CE = s − c

These tangent lengths appear frequently in geometric proofs and are essential when constructing the incircle manually.

Practical Applications

The inradius has applications beyond pure mathematics. Engineers and architects use it to determine the largest circular cross-section that fits inside a triangular structural member or aperture. In computational geometry, the inradius provides a measure of how "fat" or well-proportioned a triangle is — triangles with a larger inradius relative to their size are considered less degenerate. In numerical methods, meshes with triangles that have a large inradius tend to be better-conditioned.

Validation and Accuracy

All calculations use standard 64-bit floating-point arithmetic (IEEE 754 double precision), providing roughly 15–16 significant digits. The calculator enforces the triangle inequality for SSS mode (a + b > c for all permutations), validates that all side lengths and angles are strictly positive, and rejects degenerate triangles where the computed area is effectively zero. Results are displayed with a configurable number of decimal places (0–10).

Frequently Asked Questions

Is the Inradius of Triangle free?

Yes, Inradius of Triangle is totally free :)

Can I use the Inradius of Triangle offline?

Yes, you can install the webapp as PWA.

Is it safe to use Inradius of Triangle?

Yes, any data related to Inradius of Triangle only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is the inradius of a triangle?

The inradius r is the radius of the inscribed circle (incircle) — the largest circle that fits perfectly inside the triangle, touching all three sides. Every non-degenerate triangle has exactly one incircle, and its center is called the incenter.

What is the incircle and incenter?

The incircle is the circle tangent to all three sides of the triangle from the inside. Its center, the incenter (I), is found at the intersection of the three angle bisectors of the triangle. The distance from the incenter to each side equals the inradius r.

How does this inradius calculator work?

Enter three side lengths (SSS mode) and the calculator computes the semi-perimeter s = (a+b+c)/2 and the area via Heron's formula, then applies r = Area / s. You can also switch to SAS, AAS/ASA, or Right Triangle mode — each uses the appropriate formula to derive the full triangle before computing r. Results include the incircle circumference (2πr), incircle area (πr²), triangle type, and a step-by-step breakdown.

How is the formula r = Area / s derived?

If you drop perpendiculars from the incenter I to each side, each has length r. The triangle can be split into three smaller triangles — one for each side — each with base equal to that side and height r. Summing their areas: Area = (1/2)·a·r + (1/2)·b·r + (1/2)·c·r = r·(a+b+c)/2 = r·s. Rearranging gives r = Area / s.

How does the right triangle shortcut r = (a + b − c) / 2 work?

For a right triangle with legs a and b and hypotenuse c, the area is a·b/2 and the semi-perimeter is (a+b+c)/2. Substituting into r = Area/s gives r = (a·b/2) / ((a+b+c)/2) = a·b/(a+b+c). This simplifies to r = (a+b−c)/2 because for a right triangle, c² = a²+b², which makes the algebra work out cleanly.

What is the difference between inradius and circumradius?

The inradius r is the radius of the inscribed circle (tangent to all three sides from inside), while the circumradius R is the radius of the circumscribed circle (passing through all three vertices). For any triangle, R ≥ 2r, with equality only for an equilateral triangle. They are related by Euler's formula: OI² = R(R − 2r), where O is the circumcenter and I is the incenter.