Circumscribed Circle Calculator
Enter the three side lengths of your triangle to find the circumscribed circle (circumcircle): the unique circle that passes through all three vertices. The calculator gives you the circumradius R, the circle diameter, circumference and area, plus the triangle area and how much larger the circle is than the triangle. Switch between centimetres and inches - results update instantly.
Formula
Worked example
Triangle with sides a = 5, b = 7, c = 8 cm. Semi-perimeter s = (5+7+8)/2 = 10. Area A = sqrt(10 x 5 x 3 x 2) = sqrt(300) = 17.3205 cm^2. Circumradius R = (5 x 7 x 8) / (4 x 17.3205) = 280 / 69.282 = 4.0415 cm. Inradius r = 17.3205 / 10 = 1.7321 cm. Euler distance OI = sqrt(4.0415 x (4.0415 - 2 x 1.7321)) = sqrt(4.0415 x 0.5773) = sqrt(2.3337) = 1.5277 cm.
What is the circumscribed circle of a triangle?
The circumscribed circle, also called the circumcircle, is the unique circle that passes exactly through all three vertices of a triangle. Every triangle has exactly one circumcircle - this is guaranteed by the fact that any three non-collinear points determine a unique circle. The center of the circumcircle is called the circumcenter, denoted O, and is the point where the three perpendicular bisectors of the triangle's sides meet. The radius is called the circumradius, R. The circumradius formula is R = abc / (4A), where a, b, c are the side lengths and A is the triangle area. You can compute A from the three sides alone using Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter. For a right triangle there is an especially elegant shortcut: the circumcenter sits exactly at the midpoint of the hypotenuse, so R = hypotenuse / 2. For an equilateral triangle with side a, the formula reduces to R = a / sqrt(3).
The law of sines and the circumradius
The circumradius appears naturally in the law of sines, one of the most fundamental relationships in triangle geometry. The law states that for any triangle: a / sin(A) = b / sin(B) = c / sin(C) = 2R, where the capital letters denote the angles opposite the corresponding lowercase sides. This means you can also find R from a single side and its opposite angle: R = a / (2 sin A). This connection makes the circumradius a bridge between the metric (length-based) and trigonometric descriptions of a triangle. The circumcircle area ratio - how many times larger the circumcircle is than the triangle - is always at least pi/3 (about 1.047), achieved in the limit by equilateral triangles. For very flat (nearly degenerate) triangles the ratio grows without bound, because the circumcircle has to stretch far to catch the near-collinear vertices while the triangle area shrinks to zero.
Incircle, circumcircle and Euler's inequality
The incircle (inscribed circle) is the largest circle that fits inside the triangle, tangent to all three sides. Its radius r (the inradius) is given by r = A/s, where A is the triangle area and s the semi-perimeter. Together, R and r tell you a great deal about the triangle's shape. Euler's inequality states that R >= 2r, with equality only for equilateral triangles. Euler also proved an exact formula for the distance OI between the circumcenter O and the incenter I: OI^2 = R(R - 2r). This calculator computes both the inradius and the Euler distance as bonus outputs, so you can see how the two circles relate for any triangle you enter. For a right triangle, the circumcenter lies on the hypotenuse and the incenter lies inside, giving an Euler distance you can verify: if the legs are p and q and hypotenuse c, then r = (p + q - c)/2 and R = c/2.
How to use this calculator
Enter the three side lengths of your triangle in the boxes labeled Side a, Side b and Side c. Choose centimetres or inches from the unit selector - all outputs switch automatically. The calculator instantly gives you the circumradius R, diameter, circumference and area of the circumcircle, plus the triangle area, perimeter and area ratio. You also get the inradius and Euler OI distance as bonus results. The "show your work" panel walks through every arithmetic step with your exact numbers substituted in. The reference table at the bottom shows simplified formulas for equilateral, right and isosceles triangles. If the three lengths violate the triangle inequality (one side is not shorter than the sum of the other two), all outputs clear rather than giving a nonsense answer.
Circumradius formulas by triangle type
| Triangle type | Simplified formula | Notes |
|---|---|---|
| Equilateral (a = b = c) | R = a / sqrt(3) | Circumcenter = incenter = centroid; R = 2r |
| Right (angle C = 90 deg) | R = c / 2 | Circumcenter lies at hypotenuse midpoint |
| Isosceles (a = b) | R = a^2 c / (4A) | Circumcenter lies on axis of symmetry |
| General scalene | R = abc / (4A) | Heron's formula gives A from the three sides |
| Any triangle | R = a / (2 sin A) | Law of sines form; A is the angle opposite side a |
Special-case formulas that simplify R = abc / (4A) for common triangle types.
Frequently asked questions
What is the formula for the circumradius of a triangle?
The circumradius formula is R = abc / (4A), where a, b and c are the three side lengths and A is the area of the triangle. Area A is calculated from the sides alone using Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), with s = (a+b+c)/2 the semi-perimeter. An equivalent form via the law of sines is R = a / (2 sin A), where A is the angle opposite side a.
What is the circumscribed circle of a right triangle?
For a right triangle the circumcenter lies exactly at the midpoint of the hypotenuse. This means the circumradius equals half the hypotenuse length: R = c/2, where c is the hypotenuse. You can verify this with the general formula: in a right triangle A = 90 degrees, so sin(A) = 1, and the law of sines gives 2R = c / sin(90 deg) = c.
How is the circumscribed circle different from the inscribed circle?
The circumscribed circle (circumcircle) passes through all three vertices and lies outside the triangle. The inscribed circle (incircle) is tangent to all three sides and lies inside the triangle. The circumradius R is always at least twice the inradius r (Euler's inequality R >= 2r), with equality only for equilateral triangles. Euler's formula OI^2 = R(R - 2r) gives the exact distance between their centers.
What is the circumradius formula for an equilateral triangle?
For an equilateral triangle with side length a, all three sides are equal so a = b = c. The triangle area is A = (sqrt(3)/4) a^2 and the semi-perimeter is s = 3a/2. Substituting into R = abc/(4A) gives R = a^3 / (4 x (sqrt(3)/4) a^2) = a / sqrt(3) = a sqrt(3) / 3. The equilateral triangle also achieves the minimum area ratio (pi/3) and is the only triangle where R = 2r exactly.
What does the circumcircle area to triangle area ratio mean?
The ratio Ac/At = (pi R^2) / A tells you how much of the circumcircle's area is actually filled by the triangle. For an equilateral triangle the ratio is exactly pi/sqrt(3) (about 1.814), which is the minimum possible value - equilateral triangles are the most efficient at filling their circumcircle. For any other triangle the ratio is larger, meaning more of the circumcircle is wasted space. Very flat or obtuse triangles can have very large ratios.
Does every triangle have a circumscribed circle?
Yes - every triangle has exactly one circumscribed circle. This follows from the fact that any three non-collinear points (which is what the three vertices of a valid triangle are) lie on exactly one circle. The circumcenter is the intersection of the three perpendicular bisectors of the sides, and this point is always unique. The only case where no circumcircle exists is when the three 'vertices' are collinear, which means they don't form a triangle at all.
Where is the circumcenter located relative to the triangle?
The position of the circumcenter O depends on the triangle type. For an acute triangle (all angles less than 90 degrees), O lies inside the triangle. For a right triangle (one angle exactly 90 degrees), O lies on the hypotenuse. For an obtuse triangle (one angle greater than 90 degrees), O lies outside the triangle on the side opposite the obtuse angle. For an equilateral triangle O coincides with the incenter, centroid and orthocenter.