# What is the length of an arch whose top height is 6 foot above a 20 foot span?

End to end on this building it's 20 feet. So If I build an arch that bows to a top height of 6 feet.. What is the length of that arch? ( i know this isn't rocket science but it is to me !)

Relevance

There are several shapes arches can have. Part of a circle or part of a parabola are two that come to mind.

If I assume part of a circle... you have a circle segment that is 6 ft high and 20 ft end to end (the chord of the circle).

You want the length, by which I assume you mean the perimeter.

Circle

given a circle with height h (distance from chord to arc, with the height

centered on the chord and perpendicular to it). h = 6

and L = length of chord = 20

and r = radius of circle

r = (L²/8x) + (x/2)

r = (20²/8•6) + (6/2) = 8.33+3 = 11.33 ft

d (diameter) = 22.67 ft

central angle

θ = arcsin(L/2r) = arcsin(20/2•11.33)

θ = 61.9º

arc length s = dπ(θ/360) = 12.25 ft

but that's not right, it has to be >20. I'll recheck. i'd guess it's actually twice that or 24.5 ft.

• Angle is TWO arcsin, Bill..