Skip to main content
math

Circumscribed Circle (Circumradius) Calculator

Calculate the circumradius (radius of the circumscribed circle) of a triangle given its three sides using R = abc/(4*Area). The circumscribed circle passes through all three vertices. Used in geometry, trigonometry, navigation, and satellite positioning.

Reviewed by Christopher FloiedPublished Updated

This free online circumscribed circle (circumradius) calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Minimum: 0.01

Length of side a

Minimum: 0.01

Length of side b

Minimum: 0.01

Length of side c

Results

Circumradius (R)

2.5

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Circumscribed Circle (Circumradius) Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Circumscribed Circle (Circumradius) Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Circumscribed Circle (Circumradius) Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

Related Calculators

Scalene Triangle Area Calculator

Calculate the area of any triangle from its three side lengths using Heron's formula. Enter sides a, b, and c to compute the area without needing height or angles, perfect for surveying, land measurement, navigation, and general geometry problems.

30-60-90 Triangle Calculator

Calculate all sides of a 30-60-90 special right triangle from any one known side. Enter the shortest side, longer leg, or hypotenuse to find all three sides using the fixed ratio 1:sqrt(3):2, fundamental for trigonometry, geometry, and standardized testing.

Central Angle Calculator

Calculate the central angle of a circle given the arc length and radius using theta = arc/radius. The central angle is the angle at the center subtended by an arc, fundamental to circle geometry, surveying, navigation, and engineering applications.

Circle Area Calculator

Calculate the area of a circle from its radius using the formula A = πr². Essential for geometry, engineering, and everyday measurements.

Right Triangle Trigonometry Calculator

Solve a right triangle given two sides. Enter any two sides (legs a and b) to compute the hypotenuse, all angles, area, and trigonometric ratios using the Pythagorean theorem and inverse trigonometric functions for physics and engineering problems.

Triangle Similarity Calculator

Check if two triangles are similar by comparing their side ratios (SSS similarity test). Enter three sides of each triangle to determine if corresponding sides are proportional and find the scale factor, essential for geometry proofs and real-world scaling problems.

About Circumscribed Circle (Circumradius) Calculator

The Circumscribed Circle Calculator computes the circumradius R of a triangle, which is the radius of the unique circle passing through all three vertices (the circumscribed circle or circumcircle). The formula R = abc/(4*Area) connects the circumradius to the side lengths and area. Every triangle has exactly one circumscribed circle, and its center (the circumcenter) is the intersection of the perpendicular bisectors of the sides. The circumradius is important in trigonometry (the law of sines states a/sin(A) = 2R), navigation (three-point resection uses circumcircles), satellite positioning, and computational geometry (Delaunay triangulation maximizes the minimum angle by using circumcircles).

The Math Behind It

The circumscribed circle (circumcircle) of a triangle passes through all three vertices. Its center, the circumcenter, is equidistant from all three vertices and is located at the intersection of the perpendicular bisectors of the sides. The circumradius R can be derived from the law of sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R. Since the area of a triangle is (1/2)*a*b*sin(C), we have sin(C) = 2*Area/(a*b). Substituting: 2R = c/sin(C) = c*a*b/(2*Area), so R = abc/(4*Area). Using Heron's formula for the area, R = abc/(4*sqrt(s(s-a)(s-b)(s-c))). For a right triangle with hypotenuse c, the circumradius is simply R = c/2 (the circumcenter is the midpoint of the hypotenuse). For an equilateral triangle with side s, R = s/sqrt(3) = s*sqrt(3)/3. The circumcenter's position depends on the triangle type: for acute triangles, it lies inside the triangle; for right triangles, it is at the midpoint of the hypotenuse; for obtuse triangles, it lies outside the triangle. This geometric fact has practical implications in surveying and navigation. In computational geometry, the circumcircle is central to Delaunay triangulation, which has the property that no point lies inside any triangle's circumcircle. This produces well-shaped triangles (maximizing minimum angles) and is used in mesh generation, terrain modeling, and finite element analysis. The Voronoi diagram, dual to the Delaunay triangulation, also uses circumcircle properties.

Formula Reference

Circumradius Formula

R = abc / (4 * Area) = abc / (4 * sqrt(s(s-a)(s-b)(s-c)))

Variables: a, b, c = sides, s = semi-perimeter, Area from Heron's formula

Worked Examples

Example 1: 3-4-5 Right Triangle

Find the circumradius of a 3-4-5 right triangle.

Step 1:Area = (1/2)*3*4 = 6
Step 2:R = (3*4*5)/(4*6) = 60/24 = 2.5
Step 3:Cross-check: R = c/2 = 5/2 = 2.5 (confirmed for right triangle)

The circumradius is 2.5.

Example 2: Equilateral Triangle, side 6

Find the circumradius of an equilateral triangle with side 6.

Step 1:s = (6+6+6)/2 = 9
Step 2:Area = sqrt(9*3*3*3) = sqrt(243) = 15.588
Step 3:R = (6*6*6)/(4*15.588) = 216/62.354 = 3.464
Step 4:Cross-check: R = 6/sqrt(3) = 3.464 (confirmed)

The circumradius is approximately 3.464.

Common Mistakes & Tips

  • !Confusing circumradius (circumscribed circle, passes through vertices) with inradius (inscribed circle, tangent to sides). They are different: R = abc/(4*Area) vs r = Area/s.
  • !Forgetting that the circumcenter is outside the triangle for obtuse triangles. The formula still works correctly; only the geometric position changes.
  • !Using incorrect area values. Double-check the area computation using Heron's formula before dividing into abc.

Related Concepts

Scalene Triangle Area

Heron's formula provides the area needed for the circumradius calculation.

Inscribed Angle

The inscribed angle theorem relates angles in a circumscribed circle to the central angle and arc.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Inscribed Angle Calculator

Frequently Asked Questions

What is the difference between circumradius and inradius?

The circumradius R is the radius of the circle passing through all three vertices (circumscribed circle). The inradius r is the radius of the circle inscribed inside the triangle, tangent to all three sides. For any triangle, R >= 2r (Euler's inequality), with equality only for equilateral triangles.

Where is the circumcenter located?

The circumcenter is the intersection of the three perpendicular bisectors of the sides. It is inside the triangle for acute triangles, on the hypotenuse for right triangles, and outside the triangle for obtuse triangles.

How is the circumcircle used in GPS?

GPS positioning uses a similar principle: three satellites define three spheres, and the receiver's position is at their intersection. In 2D, the circumcircle of a triangle formed by three known points can be used for trilateration, determining the position of a point based on distances.

Embed this calculator on your site

Paste this snippet into your blog, course page, or documentation to drop a live, interactive Circumscribed Circle (Circumradius) Calculator into your page.

Free to embed — includes a link back to MegaCalc.