How to use this calculator
- Enter or choose what you know.
- The circumradius R updates as you type. Use Copy result or Share to keep it.
Method
R = abc ÷ (4K), with K from Heron’s formula; R = a ÷ (2 sin A); circumcentre = point equidistant from the three vertices
- a, b, c
- the side lengths
- K
- the triangle’s area
- A
- the angle opposite side a
Worked example
- Semi-perimeter s = (5 + 6 + 7) ÷ 2 = 9.
- Heron: K = √(9 × 4 × 3 × 2) = √216 ≈ 14.696938.
- R = (5 × 6 × 7) ÷ (4 × 14.696938) = 210 ÷ 58.787754 ≈ 3.572173.
A triangle with sides 5, 6 and 7 has a circumradius of about 3.5722, so its circumcircle is about 7.1443 across.
What the result means
For an acute triangle the circumcentre sits inside it; for a right triangle it is the midpoint of the hypotenuse; for an obtuse triangle it lies outside. One side and its opposite angle fix R but not the rest of the triangle.
Assumptions and limits
- Angles are in degrees.
- Side lengths and coordinates share one unit; R is in that unit.
- Nearly flat triangles have a very large circumradius that is sensitive to rounding in the inputs.
Questions
Is the circumradius the same as the inradius?
No. The circumradius belongs to the circle through the corners (outside the triangle). The inradius belongs to the circle touching all three sides (inside). R ≥ 2r always, with equality only for an equilateral triangle.
Why does one side and its opposite angle give R?
By the extended law of sines, a ÷ sin A = 2R for every side. Any triangle with that side and opposite angle fits the same circle.