About

Every triangle has exactly one circumscribed circle (circumcircle) passing through all three vertices. The radius R of this circle is called the circumradius, and its center - the circumcenter - is the intersection of the triangle's perpendicular bisectors. Miscalculating R in structural engineering or CNC path planning leads to material waste and dimensional errors that propagate through downstream geometry. This tool computes the circumradius and circumcenter coordinates from either three side lengths or three vertex coordinate pairs, applying Heron's formula for area and the extended law of sines internally. Results are approximate for floating-point inputs; the tool assumes a Euclidean plane and non-degenerate triangles (all three vertices non-collinear).

Formulas

Given a triangle with side lengths a, b, c, the circumradius R is derived from the area K computed via Heron's formula.

s = a + b + c2

K = √s(s − a)(s − b)(s − c)

R = a ⋅ b ⋅ c4K

For vertex coordinates A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), the circumcenter (O_x, O_y) is found by solving the perpendicular bisector equations. The area is computed via the shoelace formula:

K = 12 |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Where s = semi-perimeter, K = triangle area, R = circumradius, a, b, c = side lengths opposite vertices A, B, C respectively.

Reference Data

Triangle Type	Circumcenter Location	Circumradius Relation	Interior Angle Condition
Acute	Inside the triangle	R < longest side	All angles < 90°
Right	Midpoint of hypotenuse	R = c2	One angle = 90°
Obtuse	Outside the triangle	R > longest side ÷ 2	One angle > 90°
Equilateral	Centroid (all centers coincide)	R = a√3	All angles = 60°
Isosceles (acute)	On axis of symmetry, inside	Depends on apex angle	Apex < 90°
Isosceles (obtuse)	On axis of symmetry, outside	Depends on apex angle	Apex > 90°
Scalene (acute)	Inside, off-center	Computed via Heron + formula	All angles distinct, < 90°
Scalene (obtuse)	Outside, opposite obtuse vertex	Computed via Heron + formula	One angle > 90°

Regular Polygon	Sides (n)	Circumradius Formula	Example: side = 10
Equilateral Triangle	3	R = a√3	5.7735
Square	4	R = a√22	7.0711
Pentagon	5	R = a2 sin(π/5)	8.5065
Hexagon	6	R = a	10.0000
Octagon	8	R = a2 sin(π/8)	13.0656
Decagon	10	R = a2 sin(π/10)	16.1803
Dodecagon	12	R = a2 sin(π/12)	19.3185
General n-gon	n	R = a2 sin(π/n)	Varies

Frequently Asked Questions

When all three vertices are collinear, the area K equals 0, making the circumradius formula divide by zero. No circumscribed circle exists for a degenerate triangle. The calculator detects this condition and reports an error.

For obtuse triangles (one angle greater than 90°), the circumcenter lies outside the triangle, on the opposite side of the longest edge from the obtuse vertex. The diagram in this calculator shows this clearly by rendering the center position relative to the triangle.

JavaScript uses IEEE 754 double-precision floats with approximately 15 - 17 significant digits. For triangles with side lengths differing by more than 10 orders of magnitude, cancellation errors in Heron's formula can produce inaccurate areas. This calculator uses a numerically stable variant that sorts sides before computation, but results are displayed to 6 decimal places to reflect practical precision limits.

No. The formula R = abc / (4K) assumes a flat Euclidean plane. On a sphere (spherical geometry) or hyperbolic plane, the circumradius requires different formulations involving the curvature of the space. This tool computes Euclidean circumradii only.

The extended law of sines states asin A = 2R, where A is the angle opposite side a. This means each side divided by the sine of its opposite angle yields twice the circumradius. The calculator uses the equivalent area-based formulation because it avoids computing angles explicitly.

Compute the distance from the circumcenter to each vertex. All three distances must equal the circumradius R within rounding tolerance. The calculator performs this verification internally and displays the circumcenter coordinates along with R for manual checking.