Circumscribed Circle Calculator

Understanding the Circumscribed Circle

Every triangle has a unique circumscribed circle, also called its circumcircle. This circle touches all three vertices and encloses the entire triangle. The radius of this circle—called the circumradius—depends only on the triangle's shape and size.

The circumcircle always exists for any triangle, whether it's acute, right-angled, or obtuse. The circumcenter (the circle's center) lies inside acute triangles, on the hypotenuse of right triangles, and outside obtuse triangles. Finding the circumcenter geometrically requires drawing two perpendicular bisectors of the triangle's sides; their intersection point is the circumcenter.

This property is fundamental in surveying, navigation, and structural design, where determining circular boundaries around three fixed points appears frequently.

The Circumradius Formula

To calculate the circumradius, you need the three side lengths and the triangle's area. Heron's formula lets you find the area from the sides alone:

• a, b, c — The three side lengths of the triangle
• S — The semi-perimeter, equal to half the triangle's perimeter
• Area — The triangle's area, calculated using Heron's formula
• R — The circumradius, the radius of the circumscribed circle

Derived Circle Properties

Once you have the circumradius, calculating other properties of the circumscribed circle is straightforward:

• Diameter: Simply double the radius: d = 2R
• Circumference: Use the standard formula: C = 2πR
• Area of the circle: A = πR²
• Area ratio: Dividing the circle's area by the triangle's area shows how much larger the circle is than the triangle it encloses

The inradius (radius of the inscribed circle inside the triangle) can also be computed: r = 2 × Area / Perimeter. This is distinct from the circumradius and typically smaller.

Key Considerations When Using This Calculator

Avoid common pitfalls when working with circumscribed circles.

1. Right triangles are a special case — For any right triangle, the circumradius equals half the hypotenuse, and the circumcenter sits exactly at the midpoint of the hypotenuse. You can verify this using the general formula—it always holds for the 90° angle.
2. Equilateral triangles have simple geometry — An equilateral triangle with side length a has circumradius R = a / √3 ≈ 0.577a. The circumcenter, centroid, and orthocenter all coincide at the same point in the middle.
3. Input order doesn't affect the result — The circumradius depends only on the three side lengths, not on which one you label a, b, or c. However, ensure all sides satisfy the triangle inequality: the sum of any two sides must exceed the third.
4. Very small area values cause large radius values — If the triangle is very flat (nearly degenerate), its area approaches zero, making the circumradius extremely large. Check that your triangle sides form a valid, non-degenerate triangle before trusting extreme radius values.

Geometric Construction of the Circumcircle

To manually construct a circumscribed circle without a calculator:

1. Draw the first side of the triangle and find its perpendicular bisector (the line crossing it at 90° through its midpoint).
2. Repeat for a second side.
3. Mark the intersection of these two perpendicular bisectors—this is the circumcenter.
4. Measure the distance from this center to any vertex (this is the circumradius).
5. Draw the circle with this radius centered at your circumcenter; it will pass through all three vertices.

This compass-and-straightedge method is exact and works for any triangle. The third perpendicular bisector will always pass through the same point, confirming your construction.