The chord of an arc is a straight line joining two points of the arc. A chord divides a circle into segments (qv. sector). The formula for the chord of an arc is:

c = 2r × sin(theta/2)

Where c is the chord length, r is the radius of the arc, and theta is the angle (from the centre of a circle described by the arc) between the two chord ends.

Prove it!

  1. Draw a circle.
  2. Draw two lines from the centre to the edge. Mark them r1 and r2.
  3. Mark the angle between these lines theta.
  4. Draw a line between the points where these lines meet the edge. Label it c.
  5. Draw a line, perpendicular to c, and passing through the centre of the circle. Mark it b
  6. We now note that the angles between b and c are right, so the triangle formed by b,r1 and half of c is right.
  7. Pulling out our trigonometry (sin theta = o/a), we get sin (theta/2) = (c/2) / r1
  8. Rearranging gives us (c/2) = r1 × sin (theta/2)
  9. And leads us to c = 2r × sin(theta/2) QED

An interesting point we can find from this is the chord lengths of hexagons, which can be shown to be identical to the radius of the circle. This means that a hexagon can be drawn with only the aid of a straight edge and a pair of compasses.