Math CentralQuandaries & Queries


Question from Wayne:

Given the length & radius of an arc, is there a formula that will accurately calculate the chord length?
I'm an architectural designer, and would need it explained in layman's terms. Thank you.


I would do it in 2 steps.

Step 1:

Find the measure of the angle t in the diagram.


The length a of the arc is a fraction of the length of the circumference which is 2 π r. In fact the fraction is fraction.

The angle t is a fraction of the central angle of the circle which is 360 degrees. It's the same fraction. Thus

t = 360 × fraction degrees.


Find the length c of the chord RQ in the diagram


S is the midpoint of RQ so |SQ| = c/2 and the measure of the angle SPQ is t/2. Also QSP is a right angle so sin(t/2)= |SQ|/r. Hence |SQ| = r × sin(t/2) and thus

c = 2 × r × sin(t/2).


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