Question

How to find a circumference of the circle that circumscribes a triangle with side lenghts 3,4,5

Answer 1

   
$\displaystyle \\ \texttt{We have } \Delta ABC \texttt{ with: } \\ \\ AB = 3 \\ AC = 4 \\ BC = 5 \\ \\ AB^2 + AC^2 = 3^2+4^2 = 9 + 16 = 25 = BC^2 \\ \Longrightarrow~~ \ \textless \ A = 90^o \\ \Longrightarrow~~ BC = \texttt{hypotenuse} \\ \Longrightarrow~~ \texttt{diameter of the circle: } D = BC = 5 \\ \\ \Longrightarrow~~R = \frac{D}{2} = \frac{5}{2} = 2,5 \\ \\ \Longrightarrow~~\texttt{circumference of the circle: } = 2\pi R = \pi D = \boxed{5\pi}$