How To Calculate the Radius of the Circumscribed Circle of a Triangle Given the Lengths of the Triangle's Three Sides

• Calculation formula of the radius of the circumscribed circle of a triangle

• To calculate R, the radius of the circle circumscribing a triangle △ABC, when given the lengths of the three sides of the triangle, apply the formula below:
• 

  Circumscribed circle of a triangle, of radius R

• R = ^{(a × b × c)} / _{(4 × Area of the triangle)}
• Where:
• a, b and c are the lengths of the three sides of the triangle △ABC
• R is the radius of the circumscribed circle of the triangle △ABC
• Area is the area of the triangle △ABC
• * Area can be calculated given a, b and c / Heron's formula

» Proof of the Calculation Formula of the Radius of the Circumscribed Circle of a Triangle

• Note: In one of the examples below you'll see that you don't even need to know the Area of the triangle in order to calculate the radius R of the circumscribed circle.

• Example 1 of radius calculation:

• The lengths of the three sides of a triangle are: a = 33, b = 45, c = 38. The area of the triangle is 614. Calculate the radius of the circumscribed circle of this triangle.
• Apply the calculation formula of the radius of the cirumscribed circle of the triangle:

• Example 2 of radius calculation:

• This time we are not given the area of the triangle, but only the lengths of its three sides: a = 24, b = 27, c = 19. Calculate the radius of the circumscribed circle of this triangle.
• Before we apply the calculation formula of the radius of the cirumscribed circle, we must calculate the area of the triangle. We do that by applying Heron's formula:

• Area of the triangle² = s × (s - a) × (s - b) × (s - c)
• Where:
• Area is the area of the triangle.
• a, b and c are the lengths of the triangle's sides
• s is the semiperimeter of the triangle
• s = ^{(a + b + c)} / ₂ = 35

» Heron's Formula, a Proof Using the Pythagorean Theorem

=> Area of the triangle ≈ 221.99099080819 ≈ 221.99

Apply the calculation formula of the radius of the cirumscribed circle of the triangle: