Circumradius of Triangle

Circumradius of Triangle Calculator – Circumcircle Radius Instantly

The circumradius (R) of a triangle is the radius of the unique circle that passes through all three vertices. This circle is called the circumscribed circleor circumcircle, and its center — the circumcenter (O) — is equidistant from all three vertices. Every non-degenerate triangle has exactly one circumcircle.

The Primary Formula – SSS via Heron's Formula

When all three side lengths a, b, and c are known, the circumradius is found in two steps. First compute the triangle's area using Heron's formula, then apply the circumradius formula:

s = (a + b + c) / 2

Area = √(s × (s − a) × (s − b) × (s − c))

R = (a × b × c) / (4 × Area)

For example, with sides a = 5, b = 7, c = 8: s = 10, Area = √(10 × 5 × 3 × 2) = √300 ≈ 17.3205, and R = (5 × 7 × 8) / (4 × 17.3205) ≈ 4.0415.

Alternative Input Modes

The calculator supports five input modes so you can work with whatever information is available:

• SSS — three side lengths. Heron's formula derives the area, then R = abc / (4K).
• SAS — two sides and the included angle C. The Law of Cosines finds the third side: c² = a² + b² − 2ab·cos C, then SSS is applied.
• AAS / ASA — two angles and one known side. The Law of Sines gives a direct shortcut: R = a / (2 sin A), where A is the angle opposite side a.
• Right Triangle — enter the two legs. By Thales' theorem, the hypotenuse is a diameter of the circumcircle, so R = hypotenuse / 2.
• Area + Sides — if the area is already known, skip recomputing it and apply R = abc / (4·Area) directly. Useful when the area comes from another source.

The Right Triangle Shortcut

One of the most elegant results in triangle geometry is that, for a right triangle, the circumradius equals exactly half the hypotenuse. This is a direct consequence of Thales' theorem: any angle inscribed in a semicircle is a right angle. Therefore, if a triangle has a right angle at C, then side c (the hypotenuse) is a diameter of the circumcircle, and R = c / 2.

For a 3-4-5 right triangle: hypotenuse = 5, so R = 2.5. You can confirm that the distances from the circumcenter (midpoint of the hypotenuse) to each of the three vertices are all exactly 2.5.

Circumcenter Position: Inside, On, or Outside the Triangle

The position of the circumcenter relative to the triangle depends on the triangle type:

• Acute triangle (all angles < 90°) — circumcenter lies inside the triangle.
• Right triangle (one angle = 90°) — circumcenter lies exactly on the hypotenuse, at its midpoint.
• Obtuse triangle (one angle > 90°) — circumcenter lies outside the triangle, on the far side of the longest edge.

This is why a circumcircle is not visually "inside" every triangle — for obtuse triangles, the circle extends beyond the triangle's edges. The calculator detects triangle type automatically and labels the circumcenter position accordingly.

Outputs Provided

Beyond the circumradius R itself, the calculator reports:

• Circumference — the perimeter of the circumcircle: 2πR.
• Circumcircle Area — the area enclosed by the circumcircle: πR².
• Triangle Area — computed (or confirmed) area of the triangle.
• Semi-perimeter — half the perimeter, used in Heron's formula.
• Triangle Type — Acute, Right, or Obtuse, determined from the cosine rule applied to the largest angle.
• Circumcenter Position — inside, on, or outside the triangle.

Practical Applications

• Engineering and manufacturing — sizing circular flanges or cutouts that must pass through three anchor points on a triangular component.
• Architecture and construction — designing arcs or curved rooflines that pass through three structural nodes.
• CAD and GIS — computing the minimum bounding circle of a triangular region, used in Delaunay triangulation and spatial indexing.
• Competitive mathematics — olympiad and university-level problems frequently involve circumradius in proofs about cyclic quadrilaterals, nine-point circles, and Euler lines.
• Astronomy — circumscribed circles arise when fitting a circular orbit to three known positions of a celestial body.

Relationship to the Inradius

The inradius r is the radius of the inscribed circle (tangent to all three sides), while the circumradius R is the radius of the circumscribed circle. For any triangle, R ≥ 2r, with equality holding only for an equilateral triangle. This inequality is known as Euler's inequality, and the distance between the circumcenter O and the incenter I satisfies OI² = R(R − 2r).

Adjustable Precision and Units

All results are shown in your chosen length unit (mm, cm, m, km, in, ft, or yd). You can set the display precision from 0 to 10 decimal places. All arithmetic internally uses JavaScript's 64-bit double-precision floating-point (IEEE 754), providing approximately 15 significant digits of accuracy.