What is the length of an arc with a 30 foot base

Question

A building that is 30 ft wide, what is the length of the arc of the roof.

Answer 1

We need to know the radius of the arc. If we assume in the absence of other info that the building is 30 ft tall as well as wide and that the the line from a point halfway along the base width to the edge of the roof defines the radius of the arc then we can calculate its length. The radius is the hypotenuse of a triangle with base 15 ft and height 30 ft. By Pythagoras the hypotenuse has length 15sqrt(5) ft. The radius makes an angle of tan^-1(30/15)=tan^-1(2)=63.435 deg. The angle of the sector subtending the arc is 180-2*63.435=53.13 deg. The length of the roof arc is 53.13/360 of the circumference of the circle radius 15sqrt(5)=210.744 ft. Therefore the arc length is 53.13/360*210.744 ft=31.10ft.