Circumscribed Circle
Calculate the circumradius, diameter, and area of a triangle circumcircle from three side lengths. Free online geometry tool with formula and instant results.
About This Calculator
The Circumscribed Circle Calculator computes the circumradius (R), circumdiameter (D), and area of the circumscribed circle (circumcircle) for any triangle given its three side lengths (a, b, c). The circumscribed circle is the unique circle that passes through all three vertices of the triangle. Its center -- the circumcenter -- is the intersection point of the perpendicular bisectors of the triangle's sides.
The calculator first validates the side lengths using the triangle inequality theorem: the sum of any two sides must exceed the third. If the sides form a valid triangle, the area is computed using Heron's formula K = √s(s-a)(s-b)(s-c) where s = (a+b+c)/2. The circumradius is then R = abc / (4K), and the circumdiameter is D = 2R. The resulting circumradius represents the distance from the circumcenter to any vertex of the triangle.
This geometry tool is ideal for students learning triangle geometry, teachers preparing lesson materials, and professionals in engineering, architecture, and CAD design who need quick circumcircle calculations.
Frequently Asked Questions
What is a circumscribed circle?
A circumscribed circle (circumcircle) is a circle that passes through all three vertices of a triangle. Its center is called the circumcenter, and its radius is called the circumradius.
How do you calculate the circumradius of a triangle?
The circumradius R is calculated using the formula R = abc / (4K), where a, b, c are the side lengths and K is the area of the triangle computed using Heron's formula.
What is the formula for the circumdiameter?
The circumdiameter is simply twice the circumradius: D = 2R = abc / (2K). Every triangle has a unique circumscribed circle with a diameter defined by these formulas.
What triangle side lengths are valid?
Three side lengths form a valid triangle if and only if they satisfy the triangle inequality theorem: the sum of any two sides must be greater than the third side (a + b > c, a + c > b, b + c > a).
Can any triangle have a circumscribed circle?
Yes, every triangle has a unique circumscribed circle (circumcircle). This is true regardless of whether the triangle is acute, right, or obtuse. The circumcenter is the intersection point of the perpendicular bisectors of the sides.
Is this circumscribed circle calculator free to use?
Yes, all calculators on Calculy are completely free to use with no registration or hidden charges.
What is the relationship between the circumcircle and triangle area?
Heron's formula gives the triangle area K = s(s-a)(s-b)(s-c) where s is the semi-perimeter. The circumradius formula R = abc/(4K) connects the side lengths to the circumcircle radius through the triangle area.