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The circumscribed circle (circumcircle) of a triangle is the unique circle passing through all three vertices. The circumradius R is the radius of this circle.
Formule
R = abc/(4K) where K is triangle area
- a, b, c
- triangle side lengths (length)
- K
- triangle area (length²)
- R
- circumradius (length)
Guide étape par étape
- 1R = (a × b × c) / (4 × Area)
- 2Where Area = √(s(s−a)(s−b)(s−c)) by Heron's formula
- 3s = (a + b + c) / 2 (semi-perimeter)
- 4For a right triangle: R = hypotenuse / 2
Exemples résolus
Entrée
Triangle with sides 3, 4, 5
Résultat
R = (3×4×5)/(4×6) = 2.5
Entrée
Equilateral triangle, side = 6
Résultat
R = 6/√3 = 3.464
Questions fréquentes
Does every triangle have a unique circumcircle?
Yes, every triangle has exactly one circumcircle (and circumradius), passing through all three vertices.
For a right triangle, how do I find the circumradius?
The circumradius of a right triangle equals half the hypotenuse: R = c/2.
What is the relationship between circumradius and the law of sines?
By the extended law of sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R.
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