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.
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.
Length of side a
Length of side b
Length of side c
Results
Circumradius (R)
2.5
How to Use This Calculator
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.
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.
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.
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.
Formula Reference
Circumscribed Circle (Circumradius) Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
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.
About This Calculator
The Circumscribed Circle (Circumradius) Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. 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. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
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
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.
The circumradius is 2.5.
Example 2: Equilateral Triangle, side 6
Find the circumradius of an equilateral triangle with side 6.
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
Used in These Calculators
Calculators that build on or apply the concepts from this page:
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.