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
| Field | What to enter |
|---|---|
| Side a | Units: cm, m, mm, in, ft, yd … |
| Side b | Units: cm, m, mm, in, ft, yd … |
| Side c | Units: 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) |
|---|---|
| 2.5 | 3.5921 |
| 3.75 | 3.501 |
| 5 | 3.5722 |
| 6.25 | 3.7326 |
| 7.5 | 3.991 |
| 10 | 5.0817 |
Formula
R = abc / (4·A), A = √(s(s − a)(s − b)(s − c))
Circumference = 2πR, circle area = πR²
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.
Frequently asked questions
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.