How do you find the chord of a circle given the diameter and arc?

Answer 1

The answer is easier if angles are measured in radians. At the end, you'll get a conversion from radians to degrees.

You are given a circle with diameter d, and a chord that subtends an arc of length x. Let c be the length of the chord and write r = d/2 for the radius of the circle.

The arc subtends an angle of y = x/r radians at the centre of the circle.

(If you draw a chord and the perpendicular from the centre of the circle to the midpoint of the chord, the following trigonometry is easier to understand).

sin(y) = (c/2)/r where c is the length of the chord.

so sin(y) = c/2r

and so c = 2r*sin(y) = d*sin(y) = d*sin(x/r) = d*sin(x/2d)

Everything on the right hand side is known and so c is easily calculated.

Radians-degrees

A whole circle is 360 degrees of 2*pi radians so 1 radian = 180/pi degrees. Writing 180/pi each time is tedious and also, higher maths is all in radians anyway, so I tend to avoid working with degrees. If you are using this on a calculator, make sure the angles are being measured in radians. The default units for angles in Excel is radians.