What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5?

This triangle with side length 3,4,5 is a right triangle because 3,4 and 5 is pythagora's triplet 
so than you drawn this triangle and drawn thei height from right angle on hypotenuse so writing

sin30 = height/4

1/2 = h/4

4=2h

h = 2 

knowing the property that the radius of a circle circumscribed to a triangle is equal 2/3 part of the length of the height of this triangle 

in this case r = (2/3)*2 = 4/3

formula of circumference of a circle = 2pi*r
so in this case 
C = 2*3,14*(4/3) = 6,28*4/3 =
= 25,12/3 = 8,373

hope this iwll help you