Math CentralQuandaries & Queries


Question from Mark, a parent:

How do you determine the radius or diameter of a circle based on the following information:
1. The distance along the circle between two points is 35'-2". This creates an arch.
2. The (chord) distance between the two points is 30'-8".
3. The distance from the center of the chord (on a 90 degree) to the arch is 6'-10 3/4".


In my diagram r is the radius of the circle, c is half the chord length and h is height of the arc above the chord.

circle sector

Triangle ABC is a right triangle and hence, by Pythagoras' theorem

r2 = (r - h)2 + c2
r2 = r2 - 2rh + h2 + c2

and hence

r = (h2 + c2)/2h

I changed all your measurements to inches and got

r = (82.752 + 1842)/(2 × 82.75) = 245.943 inches

which is 20' 6".

I didn't need to use the length of the arc so I decided to check my calculations by using a radius of 245.943 inches to calculate the length of the arc. The length a of the arc is given by

a = r θ

where θ is the angle at the centre of the circle measured in radians. The angle BCA is θ/2 and sin(BCA) = c/r. Hence

angle BCA = sin-1(c/r) = sin-1(184/245.943) = 0.8453

Thus θ = 2 × 0.8453 = 1.691 radians. Hence

a = r θ = 245.943 × 1.691 = 415.769 inches.

This is 34' 8" which is not the length you have for the arc. Are you sure the arch is an arc of a circle?


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