Circumradius Calculator
Calculate the circumradius of a triangle using sides or the sine rule
Input Values
Results
Visual Diagram
Formulas:
Three sides: R = abc / (4A)
where A = √[s(s-a)(s-b)(s-c)]
Sine rule: R = a / (2sin A)
O = Circumcenter (intersection of perpendicular bisectors)
Understanding Circumradius
The circumradius is the radius of the circumscribed circle (circumcircle) that passes through all three vertices of a triangle. It's a fundamental measurement in triangle geometry with numerous mathematical and practical applications.
Calculation Methods
- Three Sides Method: R = abc / (4A) where A is calculated using Heron's formula
- Sine Rule Method: R = a / (2sin A) using any side and its opposite angle
- Extended Sine Rule: a/sin A = b/sin B = c/sin C = 2R
- For right triangles: R = c/2 where c is the hypotenuse (simplest case)
Special Cases
- Equilateral triangle: R = a/√3 where a is the side length
- Right triangle: The circumcenter is at the midpoint of the hypotenuse
- Isosceles triangle: The circumcenter lies on the altitude from the apex
- For any triangle: R ≥ 2r where r is the inradius (equality for equilateral)
Applications
- GPS and navigation triangulation systems
- Delaunay triangulation in computational geometry
- Surveying and mapping
- Astronomy and orbital mechanics
- Computer graphics and mesh generation
Frequently Asked Questions
What is the circumradius of a triangle?
The circumradius is the radius of the unique circle that passes through all three vertices of a triangle. This circle is called the circumcircle or circumscribed circle.
How do you find circumradius from three sides?
Use the formula R = abc / (4A), where a, b, c are the side lengths and A is the area. Calculate the area using Heron's formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2.
What is the sine rule for circumradius?
The extended sine rule states that a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. This means R = a / (2sin A) for any side and its opposite angle.
Where is the circumcenter located?
The circumcenter (center of the circumcircle) is at the intersection of the perpendicular bisectors of the triangle's sides. It's inside acute triangles, on the hypotenuse of right triangles, and outside obtuse triangles.
What's the circumradius of a right triangle?
For a right triangle, the circumradius equals half the hypotenuse: R = c/2. This is because the hypotenuse is a diameter of the circumcircle, making the circumcenter its midpoint.
How does circumradius relate to inradius?
For any triangle, the circumradius R is always greater than or equal to twice the inradius r: R ≥ 2r. Equality holds only for equilateral triangles, where R = 2r exactly.
What's the circumradius of an equilateral triangle?
For an equilateral triangle with side length a, the circumradius is R = a/√3 or approximately R = 0.577a. This can also be written as R = a√3/3.
Can all triangles have a circumcircle?
Yes, every triangle has exactly one circumcircle. This is a fundamental property of triangles in Euclidean geometry. However, not all polygons with more than three sides have circumcircles.