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Triangle Inradius Calculator

Find the inradius of a triangle.

How it is calculated

The tool needs the three side lengths a, b, c, all in the same length unit. It first checks that each is positive and that the triangle inequality holds strictly (the sum of any two sides must exceed the third); a flat, degenerate triangle such as 1, 2, 3 returns nothing. Then it computes the semiperimeter s = (a + b + c) / 2, the area by Heron's formula A = √(s(s−a)(s−b)(s−c)), and the inradius r = A / s.

The inradius is shown to four decimals and the area to two. The output is labelled "units" and "units²" because the tool does not know whether you used centimetres, inches or something else; r is in the unit you entered and the area in its square.

Worked example

Take the triangle with sides 13, 14 and 15. Semiperimeter: s = (13 + 14 + 15) / 2 = 21. Differences: 21 − 13 = 8, 21 − 14 = 7, 21 − 15 = 6. Area by Heron: √(21 × 8 × 7 × 6) = √7056 = 84. Inradius: r = 84 / 21 = 4.0000 units. The tool also reports the area as 84.00 units². A circle of radius 4 therefore fits exactly inside, touching all three sides.

Try this example →

Common mistakes

  • Entering sides that cannot form a triangle, such as 2, 3 and 7. The longest side must be shorter than the sum of the other two; equal is not enough either, since 1, 2, 3 is a flat line and returns nothing.
  • Mixing units between sides, for example 1.5 m and 40 cm. All three sides must be converted to one unit first, otherwise r and the area are meaningless.
  • Entering the three angles or two sides and an angle. This tool takes only the three side lengths.
  • Confusing the inradius with the circumradius. The inscribed circle touches the sides from inside; the circle through the three corners is larger and uses a different formula.
  • Dividing the area by the perimeter instead of the semiperimeter. That gives half the true inradius.

About this calculator

Every triangle has a unique inscribed circle — the largest circle that fits inside it, tangent to all three sides — and its radius follows from a simple ratio: r = Area / s, where s is the semiperimeter, s = (a + b + c) / 2. The area itself is typically found first via Heron's formula, Area = √(s(s−a)(s−b)(s−c)), from the three side lengths alone.

This relationship connects a triangle's size directly to the radius of the circle that can be cut from its interior, which is why machinists and metalworkers use it when fitting the largest circular disk or fillet inside a triangular piece of stock, and why it appears in mesh-quality measures (the ratio of inradius to circumradius) in computational geometry and CAD software.

Geometry students proving the r = Area/s relationship, engineers and designers working with triangular cross-sections, and puzzle or game designers computing inscribed shapes all rely on this formula because it links three measurable side lengths directly to an otherwise hard-to-measure interior radius.

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Comments (1)

  • Editorial team

    The inradius is the radius of the largest circle that fits inside the triangle, which is useful when you are fitting a round object into a triangular plate, bed or corner. A quick way to test the tool: a right triangle with sides 3, 4 and 5 has inradius exactly 1, which also equals (a + b − c) / 2 for any right triangle when c is the longest side. What are you trying to fit inside your triangle?

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