# Triangle Circumradius Calculator

> Find the radius of the circle through a triangle’s three corners from its three sides, one side and the opposite angle, or vertex coordinates.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/triangle-circumradius-calculator

The circumradius R is the radius of the one circle that passes through all three vertices. Every side divided by the sine of its opposite angle equals 2R.

Also called: Circumcircle calculator, Circumcentre calculator.

## Inputs
- What you know (options: Three sides, One side and the angle opposite it, Coordinates of the three vertices; default sss)
- Side a (default 5)
- Side b (default 6)
- Side c (default 7)
- Side (default 10)
- Angle opposite that side (unit °; default 30)
- Vertex 1: x (default 0)
- Vertex 1: y (default 0)
- Vertex 2: x (default 4)
- Vertex 2: y (default 0)
- Vertex 3: x (default 0)
- Vertex 3: y (default 3)

## Results
- Circumradius R
- Circumcircle diameter (2R)
- Circumcircle circumference
- Circumcircle area
- Triangle area
- Circumcentre
- Where the centre lies

## Formula

```
R = abc ÷ (4K), with K from Heron’s formula;  R = a ÷ (2 sin A);  circumcentre = point equidistant from the three vertices
```

Where:
- a, b, c: the side lengths
- K: the triangle’s area
- A: the angle opposite side a

## Worked example
1. Semi-perimeter s = (5 + 6 + 7) ÷ 2 = 9.
2. Heron: K = √(9 × 4 × 3 × 2) = √216 ≈ 14.696938.
3. R = (5 × 6 × 7) ÷ (4 × 14.696938) = 210 ÷ 58.787754 ≈ 3.572173.

**Result:** 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.

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- [Triangle Calculator](https://calculators.geekmidst.com/calculators/triangle-calculator)
- [Right Triangle Calculator](https://calculators.geekmidst.com/calculators/right-triangle-calculator)
- [Law of Sines Calculator](https://calculators.geekmidst.com/calculators/triangle-law-of-sines-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/triangle-circumradius-calculator