Last updated October 10, 2026 · Report an error
The circumscribed circle (or circumcircle) of a triangle is the unique circle that passes through all three of its vertices — every triangle, no matter its shape, has exactly one. Its radius, called the circumradius, comes from the elegant formula R = abc / (4K), where a, b, and c are the triangle's three side lengths and K is its area, linking a triangle's size and shape directly to the circle that perfectly circumscribes it. This relationship is useful well beyond pure geometry homework — it appears in triangulation and surveying (finding a circle through three known reference points), in design work involving circular boundaries, and it's a standard building block in more advanced geometry involving the Law of Sines, since the circumradius relates directly to each side and its opposite angle. If exact vertex coordinates are given rather than just side lengths, this tool also finds the circumcircle's exact center point, called the circumcenter.
How it works
Provide either three side lengths, or three vertex coordinates. With side lengths only, the calculator first finds the triangle's area K using Heron's formula (which computes area purely from the three side lengths), then applies R = abc / (4K) to get the circumradius. With vertex coordinates provided, the calculator additionally solves for the circumcenter — the exact (x,y) point equidistant from all three vertices — by finding where the perpendicular bisectors of the triangle's sides intersect, since every point on a perpendicular bisector of a side is equidistant from that side's two endpoints.
- Enter given.
- Enter side a.
- Enter side b.
- Enter side c.
- Enter a: x₁.
- Enter a: y₁.
- Enter b: x₂.
- Enter b: y₂.
- Enter c: x₃.
- Enter c: y₃.
- Click Calculate to see your results.
Examples
Right triangle with legs 4 and 3
Vertices (0,0), (4,0), (0,3): the circumradius is 2.5 (exactly half the hypotenuse's length of 5, as is always true for any right triangle), and the circumcenter is (2, 1.5), which is precisely the hypotenuse's midpoint.
An equilateral triangle from side lengths only
For an equilateral triangle with all sides equal to 6, Heron's formula gives an area of 9√3, and R = (6×6×6)/(4×9√3) = 216/(36√3) = 6/√3 ≈ 3.464 — since only side lengths were given, no circumcenter coordinate can be reported, only the radius.
A scalene triangle from vertex coordinates
For vertices (0,0), (6,0), and (2,5), the calculator computes the side lengths, finds the area, derives the circumradius, and also solves for the exact circumcenter coordinate via the perpendicular bisector intersection — something not possible from side lengths alone.
Who should use it
- Geometry homework involving a triangle's circumscribed circle.
- Surveying or triangulation problems needing a circle through three known points.
- Verifying the Law of Sines relationship between sides, angles, and circumradius.
Industry applications
- Geometry education
- Surveying and triangulation
- Engineering and CAD design
Advantages
- Works from either side lengths or vertex coordinates.
- Finds the exact circumcenter when coordinates are available.
- Uses exact formulas (Heron's formula and perpendicular bisector intersection) rather than approximation.
Limitations
- Cannot report a circumcenter coordinate when only side lengths are given, since position information is missing.
Common mistakes to avoid
- Entering three collinear points, which don't form a triangle and have no circumscribed circle.
- Expecting a circumcenter coordinate when only side lengths (not vertex coordinates) were provided.
- Forgetting that an obtuse triangle's circumcenter falls outside the triangle itself.
Best practices
- Provide vertex coordinates (not just side lengths) whenever you need the exact circumcenter location, not just the radius.
- Use the right-triangle shortcut (circumradius = half the hypotenuse) as a quick sanity check whenever applicable.
- Verify your three points aren't collinear before expecting a valid circumscribed circle.
Tips
- For a right triangle, skip the full calculation — the circumradius is always exactly half the hypotenuse.
- Provide vertex coordinates rather than just side lengths whenever you need the circumcenter's exact location.