Circumscribed Circle Calculator

Last updated: 2026-05-20

Definition

The circumscribed circle (or circumcircle) of a triangle is the unique circle that passes through all three vertices. Its radius $R$ is called the circumradius, and its center is called the circumcenter. This calculator finds $R$, the circumcircle area, and circumference given only the three side lengths $a$, $b$, $c$.

Every triangle has exactly one circumscribed circle, unlike a general quadrilateral. The circumcircle appears throughout classical geometry and trigonometry.

Calculating the circumradius

The standard formula combines Heron's formula for the triangle area with the product of the sides:

Step 1 — Semi-perimeter:

$s = \frac{a + b + c}{2}$

Step 2 — Triangle area (Heron's formula):

$S = \sqrt{s(s-a)(s-b)(s-c)}$

Step 3 — Circumradius:

$R = \frac{abc}{4S}$

Worked example — the 7-8-9 triangle:

$s = \frac{7+8+9}{2} = 12$

$S = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx 26.83$

$R = \frac{7 \times 8 \times 9}{4 \times 26.83} = \frac{504}{107.3} \approx 4.70$

The circumcircle area is then $\pi R^2 \approx 69.4$ and the circumference $2\pi R \approx 29.5$.

The circumcenter

The circumcenter $O$ is the point equidistant from all three vertices. Geometrically, it lies at the intersection of the perpendicular bisectors of the three sides. Its position relative to the triangle depends on the triangle type:

The right-triangle case is a useful check: for sides 3, 4, 5, the circumcenter sits at the midpoint of side 5, giving $R = 5/2 = 2.5$ — which matches the formula $R = abc/(4S) = 60/24 = 2.5$.

Connection to the law of sines

The circumradius connects to the angles through the extended law of sines:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

So $R$ is half the ratio of any side to the sine of its opposite angle. This also explains the right-triangle result: when $C = 90°$, $\sin C = 1$, so $2R = c$, meaning $R = c/2$.

Circumscribed circle vs. inscribed circle

The inscribed circle (incircle) sits inside the triangle and touches all three sides. Its radius $r$ is:

$r = \frac{S}{s}$

The two radii satisfy the inequality $R \geq 2r$: the circumradius is always at least twice the inradius, with the ratio reaching its minimum of exactly 2 only for equilateral triangles. For a 3-4-5 triangle: $R = 2.5$, $r = 6/6 = 1$, so $R/r = 2.5$ — above the minimum of 2.

Both radii depend on the same triangle area $S$ and semi-perimeter $s$, but they encode different geometric properties: $R$ relates to the outer reach of the triangle, and $r$ to how much room the triangle has inside itself.

Triangle inequality

The formula only produces a real result when the three sides form a valid triangle. The triangle inequality requires:

$a + b > c, \quad b + c > a, \quad a + c > b$

If any condition fails, no triangle exists. For example, sides 1, 2, and 10 fail because $1 + 2 = 3 < 10$ — and Heron's formula would yield a negative value under the square root, making $R$ undefined.