Circumscribed Circle Radius Calculator

Work out circumscribed circle radius instantly with clear inputs, formula shown and shareable results.

About Circumscribed Circle Radius Calculator

The circumcircle passes through all three vertices. Combining the extended law of sines with Heron's area gives R = abc / (4·Area).

Formulas

Circumradius

R = abc / (4·Area)

Law of sines

a / sin A = 2R

Worked example

With the default values below, Circumscribed Circle Radius Calculator returns Circumradius R of 2.5000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Side aminimum 0: 3
Side bminimum 0: 4
Side cminimum 0: 5

Results

Circumradius R: 2.5000
Triangle area: 6.0000
Area of the circumcircle: 19.6350
Inradius for comparison: 1.0000

Step by step

1Values used
2Circumradius
3Law of sines
4Circumradius R= 2.5000
5Triangle area= 6.0000
6Area of the circumcircle= 19.6350
7Inradius for comparison= 1.0000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For the 3-4-5 triangle?

R = 60 / 24 = 2.5, exactly half the hypotenuse, as Thales' theorem requires.

Is R always at least 2r?

Yes. Euler's inequality R ≥ 2r holds for every triangle, with equality only for the equilateral case.