# Circumscribed Circle Calculator

URL: https://speedycalculator.com/calculators/math/circumscribed-circle/
Last updated: 2026-10-30

> The circumscribed circle calculator finds the circumradius, diameter, circumference and area of the circle passing through the three vertices of a triangle, from its three sides.

## Inputs

- Side a (cm, m, mm, in, ft, yd, km, mi)
- Side b (cm, m, mm, in, ft, yd, km, mi)
- Side c (cm, m, mm, in, ft, yd, km, mi)

## Results

- Triangle area
- Circumradius R
- Diameter
- Circumference
- Circle area
- Triangle area / circle area

## Formula

```
R = abc / (4·A),  A = √(s(s − a)(s − b)(s − c))
Circumference = 2πR,  circle area = πR²
```

## Explanation

Every triangle has exactly one circle through its three corners – the circumcircle. Its centre is where the perpendicular bisectors meet. It answers practical questions like the smallest round table top that fits a triangular piece or the radius of an arch through three points.

### How to use the Circumscribed Circle Calculator

- **Side a**
- **Side b**
- **Side c**

Results update as you type, and everything is calculated in your browser.

### How it works

Enter the three sides. The triangle area comes from Heron's formula and the circumradius from R = abc/(4A). The ratio shows how much of the circle the triangle fills.

### Inputs explained

FieldWhat to enterSide aUnits: cm, m, mm, in, ft, yd …Side bUnits: cm, m, mm, in, ft, yd …Side cUnits: cm, m, mm, in, ft, yd …

### What each result means

- **Triangle area** = sqrt(([Side a] + [Side b] + [Side c])×(-[Side a] + [Side b] + [Side c])×([Side a] - [Side b] + [Side c])×([Side a] + [Side b] - [Side c]))÷4
- **Circumradius R** = [Side a]×[Side b]×[Side c]÷(4×[Triangle area])
- **Diameter** = 2×[Circumradius R]
- **Circumference** = 2×pi×[Circumradius R]
- **Circle area** = pi×[Circumradius R]^2
- **Triangle area / circle area** = [Triangle area]÷[Circle area]×100

### Example values

How the main result changes when **Side a** changes and everything else stays at the example values above:

Side a (cm)Circumradius R (cm)2.53.59213.753.50153.57226.253.73267.53.991105.0817

## Worked example

Inputs — Side a: 5 cm; Side b: 6 cm; Side c: 7 cm.

Results — Triangle area = 14.696938 cm²; **Circumradius R = 3.572173 cm**; Diameter = 7.144345 cm; Circumference = 22.444622 cm; Circle area = 40.088031 cm²; Triangle area / circle area = 36.6617 %.

## Key terms

- **Circumcircle**: The circle through all three vertices.

## FAQ

**Where is the circumcenter?**
At the intersection of the perpendicular bisectors of the sides.

**Is it always inside the triangle?**
Only for acute triangles; on the hypotenuse for right triangles; outside for obtuse ones.

**What is R for a right triangle?**
Half the hypotenuse.

**How is R related to the law of sines?**
a/sin α = 2R.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [Wolfram MathWorld – Circumcircle](https://mathworld.wolfram.com/Circumcircle.html)