Math CentralQuandaries & Queries


Question from Roger:

Given the length of a chord of a circle and the height of the arc, how do I
find the lengths of equally spaced parallel lines drawn from the chord to
the arc. (Think of the lengths of vertical slats in an arch-topped bed head

We have two responses for you


One approach is to use coordinates. Let the chord have length 2c and the height of the arc be h. With the x-axis along the chord and the origin O = (0, 0) in the middle, your problem is reduced to finding the equation of the circle through the points C = (c, 0), H = (0, h), and C' = (-c, 0). Pythagoras tells you that CH has length sqrt(c2 + h2). Because the triangle HCO is similar to the triangle with vertices at H, the centre of the circle, and the midpoint of chord CH, you can use the equal ratios of the sides to obtain

radius = (h2 + c2)/(2h),

and from that you can determine that the centre is at (0, h-radius) = (0, (h2 - c2)/(2h)).

From this you get the equation of the circle. You want it in the form

y= sqrt(r2 - x2) + p,

where r is the radius and p is the y-coordinate of the centre.

If (x,0) is the bottom of the slat, plug that value of x into the equation and y will give you the length of the slat.



Hi Roger.

Take a look at a similar question I answered in 2007.

Stephen La Rocque

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