Circumscribed Circle Calculator — Circumradius of a Triangle
Our circumscribed circle calculator will become your best friend if you deal a lot with circumcircles, i.e., circles circumscribed about a triangle. Just enter the three side lengths a, b, and c, pick your units, and the calculator instantly returns the circumradius (R), diameter, circumference and area of the circumscribed circle, plus the triangle's area, perimeter, and three interior angles. Read on to learn more about this important geometric topic!
What is a circumscribed circle?
The circumscribed circle (a.k.a. the circumcircle) of a given triangle is a circle that passes through all three vertices of this triangle.
The radius of the circumcircle is called the circumradius of the triangle, and the center of the circumcircle is called the circumcenter. The circumcenter coincides with the point where the perpendicular bisectors of the triangle's sides intersect.
💡 Every triangle has a circumcircle. However, this is not true for all other polygons! For instance, among quadrilaterals, all rectangles (including squares, of course) have circumscribed circles, but no non-square rhombus has a circumscribed circle.
Formula for the circumradius of a triangle
The formula for the radius of the circumcircle of a triangle with sides a, b, and c reads:
R = (a · b · c) / (4 · Area)
where Area is the area of the triangle, which we compute from the three sides using Heron's formula:
Area = √(s(s − a)(s − b)(s − c)), s = (a + b + c) / 2
Other formulas related to circumcircles
Once you know the circumradius R, several related quantities follow immediately:
- Diameter: D = 2R
- Circumference of the circle: C = 2πR
- Area of the circle: Acircle = πR²
The circumradius can also be expressed with the law of sines, relating each side to the sine of its opposite angle:
R = a / (2·sin A) = b / (2·sin B) = c / (2·sin C)
How to use this circumscribed circle calculator?
- Select your measurement system — Metric (mm, cm, m, km) or US Customary (in, ft, yd, mi).
- Choose the unit that matches your measurements.
- Enter the three side lengths — sides a, b, and c of your triangle.
- Read off the circumradius and all related results.
The calculator verifies the triangle inequality (the sum of any two sides must exceed the third) and returns:
- Circumradius (R) and diameter (D)
- Circumference and area of the circumscribed circle
- Triangle area and perimeter
- All three interior angles
How do I circumscribe a circle about a triangle?
To construct the circumscribed circle of a triangle with a compass and straightedge:
- Draw the perpendicular bisector of any two of the triangle's sides.
- Their intersection point is the circumcenter — call it O.
- Set your compass to the distance from O to any vertex; this distance is the circumradius R.
- Draw the circle centered at O with radius R — it will pass through all three vertices.
Frequently Asked Questions
What is the circumscribed circle of a triangle?
It is the unique circle passing through all three vertices of the triangle. Its radius is the circumradius R, and its center is the circumcenter, found where the perpendicular bisectors of the sides meet.
Does every triangle have a circumscribed circle?
Yes. Every triangle — scalene, isosceles, equilateral, acute, right, or obtuse — has exactly one circumscribed circle. For a right triangle, the circumcenter lies at the midpoint of the hypotenuse, and the circumradius equals half the hypotenuse.
How do I calculate the circumradius from the three sides?
Use R = (a·b·c) / (4·Area), where the triangle's area is obtained from Heron's formula. That is exactly what this calculator does for you.
What is the difference between the circumcircle and the incircle?
The circumcircle passes through the vertices and encloses the triangle, while the incircle is the largest circle that fits inside the triangle and touches all three sides. Their radii (R and r) are generally different.
What if my sides don't form a valid triangle?
If the triangle inequality is violated (e.g., a + b ≤ c), no triangle exists, so the calculator will display an error message. A valid triangle requires a + b > c, a + c > b, and b + c > a.