Free · Triangle calculators 🔺

Triangle Circumcircle, Inradius & Area

Calculate a triangle’s circumscribed circle and related geometry from three valid side lengths.

  • Formula & worked example
  • Private in your browser
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

A 3–4–5 triangle has area 6, perimeter 12, circumradius 2.5 and inradius 1 in consistent length units. Its circumcircle diameter is the hypotenuse, 5.

Triangle area Δ = √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2; circumradius R = abc/(4Δ); inradius r = Δ/s

Worked example

Enter these known values and leave the other values blank.

Side a
3 cm
Side b
4 cm
Side c
5 cm
Circumcircle radius
2.5 cm
Circumcircle diameter
5 cm
Circumcircle circumference
15.708 cm
Circumcircle area
19.635 cm²
Triangle perimeter
12 cm
Triangle area
6 cm²
Circumcircle area / triangle area
3.2725
Triangle inradius
1 cm

Assumptions and limitations

  • All sides must be positive and satisfy every strict triangle inequality. A degenerate triangle has zero area and no finite circumcircle in this model.
  • The circumscribed circle passes through all three vertices. Its center can lie outside an obtuse triangle; the radius formula remains valid.
  • The inradius describes a different circle tangent to the three sides. It should not be confused with the circumradius.
  • Outputs are derived from the three sides and are read-only. Select units consistently; each field can display another supported unit.
  • The optional soccer-field area comparison is a fixed 7,140 m² reference, not a claim that every soccer field has that area.

Common questions

Is the circumradius always half the longest side?

That relationship holds for a right triangle, where the hypotenuse is a diameter. Other triangle shapes generally have a different relationship.

Why is a 1–2–3 triangle rejected?

Two sides sum to the third, so the vertices are collinear. The area is zero and the circumradius expression would divide by zero.

References

The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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