Circumscribed Circle of a Triangle Calculator

What is a Circumscribed Circle of a Triangle?

The circumscribed circle of a triangle is the circle that passes through all three vertices of the triangle. The center of this circle is called the circumcenter, and the radius is called the circumradius.

Formulas for Circumscribed Circle of a Triangle

Let \(A\), \(B\), and \(C\) be the sides of the triangle, and \(s\) be the semi-perimeter. Then:

1. Semi-Perimeter: \(s = \frac{A + B + C}{2}\)
2. Radius: \(r = \frac{A \times B \times C}{4 \times \sqrt{s \times (s - A) \times (s - B) \times (s - C)}}\)
3. Center: \(x = \frac{B^2 + C^2 - A^2}{2 \times B}, y = \frac{A^2 + C^2 - B^2}{2 \times A}\)

Step-by-Step Calculations

Let's calculate the radius and center of the circumscribed circle for a triangle with sides \(A = 3\), \(B = 4\), and \(C = 5\):

1. Calculate the semi-perimeter: \[s = \frac{3 + 4 + 5}{2} = 6\]
2. Calculate the radius: \[r = \frac{3 \times 4 \times 5}{4 \times \sqrt{6 \times (6 - 3) \times (6 - 4) \times (6 - 5)}} = 2.4\]
3. Calculate the center: \[x = \frac{4^2 + 5^2 - 3^2}{2 \times 4} = 3, y = \frac{3^2 + 5^2 - 4^2}{2 \times 3} = 4\]