Calculate the circumscribed circle (circumcircle) of a triangle or regular polygon. Find the circumradius, circle area, area ratios, and perimeter comparisons instantly.
The circumscribed circle — also called a circumcircle — is the unique circle that passes through all vertices of a polygon. For any triangle, a circumcircle always exists and is unique, with its center (the circumcenter) equidistant from all three vertices. For regular polygons, the circumscribed circle is the circle that touches every vertex of the polygon.
This calculator handles two modes: triangles defined by three side lengths and regular polygons defined by their number of sides and side length. For triangles, the circumradius R is calculated using the elegant formula R = abc / (4K), where a, b, c are the side lengths and K is the area found via Heron's formula. For regular polygons, R = s / (2 sin(π/n)), where s is the side length and n is the number of sides.
Understanding circumscribed circles is essential in computational geometry, CAD design, mesh generation for finite element analysis, and many areas of mathematics. The ratio between the circumscribed circle's area and the inscribed polygon reveals important insights about geometric efficiency and how polygons approximate circles as the number of sides increases. Use this calculator to explore circumradii, compare areas and perimeters, and visualize the relationship between shapes and their circumscribed circles.
This circumscribed circle calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Mode, Side a, Side b, Side c and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Triangle: R = abc / (4K), where K = √(s(s−a)(s−b)(s−c)) and s = (a+b+c)/2 Regular Polygon: R = s / (2 sin(π/n)) Circle Area = πR² Polygon Area = ½ n s² / tan(π/n)
Result: For mode=5, sidea=10, sideb=15, the tool returns the solved circumscribed circle outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circumscribed circle formulas and reports derived values, checks, and classifications automatically.
This page is tailored to circumscribed circle, with outputs tied directly to the form fields (Mode, Side a, Side b, Side c). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
A circumscribed circle (circumcircle) is the smallest circle that passes through all vertices of a polygon. Every triangle has a unique circumcircle, and every regular polygon has one as well.
Use the formula R = abc / (4K), where a, b, c are the side lengths and K is the triangle's area. The area can be found using Heron's formula.
No. All triangles and all regular polygons have circumscribed circles, but irregular polygons with 4+ sides may not. A polygon that does is called cyclic.
The circumcenter is the intersection of the perpendicular bisectors of the sides. For acute triangles it is inside, for right triangles it is on the hypotenuse midpoint, and for obtuse triangles it is outside.
For a regular hexagon with side length s, the circumradius R equals s. This is because sin(π/6) = 0.5, so R = s / (2 × 0.5) = s.
As the number of sides n increases, the regular polygon fills more of its circumscribed circle. The ratio circle area / polygon area approaches 1 as n → ∞.