Circumscribed Circle Calculator
Calculate the circumscribed circle (circumcircle) radius for any triangle passing through all vertices
Triangle Sides
Results
Visual Diagram
Formula:
R = abc / (4A)
where A = √[s(s-a)(s-b)(s-c)]
and s = (a + b + c) / 2
About Circumscribed Circles
A circumscribed circle (circumcircle) is a circle that passes through all three vertices of a triangle. The center of the circumcircle is called the circumcenter, which is the point where the perpendicular bisectors of the triangle's sides intersect.
Key Properties
- The circumcircle passes through all three vertices of the triangle
- The circumcenter is equidistant from all three vertices
- This distance equals the circumradius (R)
- The circumradius formula: R = abc / (4A), where A is the triangle's area
- Every triangle has exactly one circumscribed circle
- For a right triangle, the circumcenter lies on the hypotenuse
Applications
- Navigation and GPS triangulation
- Surveying and land measurement
- Computer graphics and computational geometry
- Architecture and structural design
- Robotics path planning
- Astronomy and celestial mechanics
Frequently Asked Questions
What is a circumscribed circle?
A circumscribed circle (or circumcircle) is a circle that passes through all three vertices of a triangle. It's the smallest circle that can contain the entire triangle.
How do you calculate the circumradius?
The circumradius (R) is calculated using the formula R = abc / (4A), where a, b, and c are the side lengths and A is the triangle's area found using Heron's formula.
Where is the circumcenter located?
The circumcenter is located at the intersection of the perpendicular bisectors of the triangle's sides. For acute triangles it's inside, for right triangles it's on the hypotenuse, and for obtuse triangles it's outside the triangle.
Does every triangle have a circumscribed circle?
Yes, every triangle has exactly one circumscribed circle. This is a fundamental property in Euclidean geometry. However, not all polygons have circumscribed circles.
What's the difference between circumradius and inradius?
The circumradius (R) is the radius of the circle passing through all vertices (outside or containing the triangle), while the inradius (r) is the radius of the circle inside the triangle touching all sides. The circumradius is always larger than the inradius.
How is the circumradius related to the sine rule?
The extended sine rule states that a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius. This relationship is useful for solving triangles when angles are known.
What happens to the circumradius in special triangles?
For an equilateral triangle with side a, R = a/√3. For a right triangle, R equals half the hypotenuse (R = c/2), making the hypotenuse a diameter of the circumcircle.
Can you construct a circumcircle with compass and straightedge?
Yes, you can construct a circumcircle by drawing the perpendicular bisectors of any two sides. Their intersection is the circumcenter. Then use this point as the center and the distance to any vertex as the radius to draw the circle.