How To Prove the Calculation Formula of the Radius of the Circumscribed Circle of a Triangle?

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

• To calculate R, the radius of the circle circumscribing the triangle △ABC, apply the formula:
• 

  Circumscribed circle of a triangle, of radius R

• R = ^{(a × b × c)} / _{(4 × Area)}
• 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

» How To Find the Circumcenter, the Center of the Circumscribed Circle of a Triangle

• How to prove the calculation formula of the radius of the circumscribed circle?

• We use the Law of sines and the Trigonometric formula of the area of a triangle to prove the formula of the radius above.
• 1) The Law of sines establishes the relationship between the lengths of the sides a, b and c of the triangle △ABC, and the sines of the opposite angles, ∠A, ∠B and ∠C:

• [1] ^a / _sin(∠A) = ^b / _sin(∠B) = ^c / _sin(∠C) = 2 × R
• Where:
• R is the radius of the circumscribed circle of the triangle △ABC
• a, b and c are the lengths of the three sides of the triangle △ABC
• ∠A, ∠B and ∠C are the angles of the triangle △ABC

» Law of Sines in a Triangle, Proof on a Circumcircle

• 2) The Trigonometric formula of the area states that the area of ​​a triangle is equal to half the product of the lengths of two sides of the triangle and the sine of the angle between them:
• 

  How to prove the calculation formula of the radius of the circumscribed circle?

• [2] Area = ¹/₂ × b × c × sin(∠A)
• Where:
• b and c are the lengths of two of the sides of triangle △ABC
• ∠A is the angle between the sides with lengths b and c

» How To Calculate Triangle's Area, Given the Lengths of Two Sides and the Measure of the Angle Inbetween. Trigonometric Formula

From the relationship [1], of the Law of sines, we can deduce the formula for the sine of angle ∠A:

[3] sin(∠A) = ^a/_{(2 × R)}

Substitute for sin(∠A) in the relationship [2], of the Trigonometric formula of the area:

Solve for R in the relationship above, of the area, and thus the calculation formula of the radius of the circumscribed circle of a triangle is proved:

R = ^{a × b × c}/_{(4 × Area)}