Circumscribed Circle Calculator - R = abc / 4K from Three Sides

A circumscribed circle calculator is a geometry tool that finds the radius, diameter, circumference, and area of the single circle that passes through the three vertices of a triangle when you enter the three side lengths. Use it whenever a homework problem, a sketch, or a construction layout gives you the three sides of a triangle and asks for the size of the circle that contains it.

• • Solve textbook problems: Find the circumradius, diameter, and circle area for a triangle in an algebra or geometry exercise.
• • Check 3-4-5 and other right triangles: Confirm the well-known result that the circumradius of a right triangle equals half the hypotenuse.
• • Plan a circular enclosure: Estimate the smallest round boundary that can contain three posts placed at the corners of a triangular footprint.
• • Compare triangle shapes: See how the circumradius changes between equilateral, isosceles, and scalene triangles with similar perimeters.

Every non-degenerate triangle has exactly one circle that passes through all three of its vertices, and that circle is called the circumscribed circle or circumcircle. The center of the circle is the circumcenter, and the radius is the circumradius R. The three vertices sit on the circle's edge, which is what makes the circle 'circumscribed' about the triangle.

The equilateral case is the most symmetric: the circumcenter sits at the centroid, and the circumradius is a / sqrt(3). For a right triangle the circumcenter sits at the midpoint of the hypotenuse, so the circumradius is half the hypotenuse. Scalene triangles land somewhere in between, and the formula below works for all three shapes.

If you only need radius, diameter, area, and circumference from a known radius, the Circle Calculator handles the arithmetic directly.

The calculator takes the three side lengths a, b, c, runs them through Heron's formula to get the triangle's area K, and then divides the product a*b*c by 4*K to land on the circumradius R. Diameter, circumference, and the circle's area are then derived from R using the standard circle formulas.

• a, b, c: Lengths of the three sides of the triangle. Each must be greater than zero and the three must satisfy the triangle inequality.
• s: Semi-perimeter, s = (a + b + c) / 2. Used as the base of Heron's formula.
• K: Area of the triangle from Heron's formula. Required to keep R dimensionally consistent.
• R: Circumradius, the radius of the circle that passes through all three vertices.

The Heron area step matters: it lets the formula absorb any triangle, not just right triangles. The product a*b*c in the numerator carries the side lengths, and 4*K in the denominator normalizes the result by the area so the units work out as a length. All results update as you type, and the units stay the same as the side units you entered.

According to Wolfram MathWorld, the circumradius of a triangle with sides a, b, c and area K is R = abc / (4K)

According to Cuemath, Heron's formula gives the area of a triangle as sqrt(s(s - a)(s - b)(s - c))

For other triangle workflows that start from angles or from two sides and an included angle, the Triangle Calculator gives the matching sides, area, and circumradius in one place.

Four ideas come up every time a circumscribed circle is computed. They are also the reason the same R = abc / 4K formula handles every triangle.

These four concepts are what make the formula portable. A right triangle, an obtuse triangle, and an acute triangle all use the same R = abc / 4K, and the difference is just where the circumcenter lands: inside an acute triangle, on the hypotenuse midpoint of a right triangle, and outside an obtuse triangle.

When the input is three coordinates or three angle-side pairs, the ABC Triangle Calculator extends the same three-side setup to the rest of the triangle.

Type the three side lengths, watch the radius, diameter, circumference, and circle area update, and read off the Heron area if you need it for a check.

1. 1 Enter side a: Type the length of the first side of the triangle into the a field. Use the same unit you want the result in.
2. 2 Enter side b: Type the length of the second side into the b field. Keep the unit consistent with side a.
3. 3 Enter side c: Type the length of the third side into the c field. The three sides must satisfy the triangle inequality.
4. 4 Read the circumradius and circle measurements: The results panel shows R, the diameter 2R, the circumference 2πR, the circle area πR², the semi-perimeter s, and the triangle area K.
5. 5 Adjust the inputs to compare triangles: Change one or more sides to see how R responds. Watch the difference between an equilateral, a right, and an obtuse triangle for similar perimeters.

A geometry problem gives sides 7, 9, 12. Type 7 into a, 9 into b, 12 into c, and the results panel shows R ≈ 6.037, diameter ≈ 12.075, circumference ≈ 37.934, and circle area ≈ 114.511.

If your three sides happen to satisfy the Pythagorean theorem, the Right Triangle Calculator gives the right-angle context alongside the circumradius.

What the calculator returns and how each output shortens a common geometry workflow.

• • One-step R from three sides: Avoid running Heron's formula and the R = abc / 4K division by hand. The calculator does both as you type.
• • Full circle at once: R, diameter, circumference, and circle area are all reported in the same unit, so a sketch or a spec sheet is ready to use.
• • Cross-checks in the same view: The semi-perimeter and the Heron area are shown alongside R, which makes it straightforward to confirm the R by recomputing it from the displayed K.
• • Triangle-inequality guard: If the three sides do not form a valid triangle, the page flags the input rather than returning a negative or imaginary radius.
• • Works for any triangle: Acute, right, and obtuse triangles are all handled by the same formula, with the circumcenter landing wherever the geometry puts it.

A student gets the circumradius, the supporting Heron area, and the rest of the circle measurements in one go. A designer working with a triangular plot can read the diameter and circle area directly from the results panel instead of pulling out a separate area tool.

If only the area is needed and not the circumscribed circle, the Triangle Area Calculator returns the Heron area and base-height area together.

A few characteristics of the input triangle change the circumradius. Knowing them tells you when to trust the result and when to double-check the inputs.

• • The calculator accepts only side-length inputs, not angle-side pairs or coordinate triples. For a triangle given as one side and the opposite angle, compute R = a / (2 sin A) by hand first.
• • Results are rounded to three decimal places. For an exam answer that needs an exact value, use R = abc / (4K) symbolically and simplify with the Heron area.

When the same three vertices are given as (x, y) points rather than as side lengths, the side lengths are pulled from the coordinates first, and the rest of the workflow is the same.

According to Math Open Reference, Heron's formula is the general triangle area form that does not need the height

When the triangle is given as three points on a coordinate plane, the Area of a Triangle by Coordinates Calculator pulls the side lengths from the coordinates before any circumradius step.