Math CentralQuandaries & Queries


Question from Richard:

I am trying to build an arched opening for a large doorway...I know the vertical sides of the opening to be 8'-9" from the floor to the lowest point of the arch on each side...I know it is 15 1/2" from the center horizontal point to the top of the arc...I know the vertical sides are 11'-11" apart...what I need to know is the radius to create the proper arc. Can you help?

Hi Richard. You can see the mathematical method I am using by reading
which solves even a bit more than you are asking, and that extra may be useful to you in your carpentry.

The short answer (based on the notation in that web page) is:

x0 = 143/2 = 71.5 inches
y0 = (15.5 - 71.52/15.5) / 2 = -157.16 inches
r2 = 71.52 + (-157.16)2 = 29811.9 inches squared

So the radius r = 172.7 inches or about 14' 4.7".

If you pick one side of the doorway and measure horizontally from there, you can determine the height that the arch should be above that location (in inches) by calculating:

h = -157.16 + sqrt(29811.9 - (x - 71.5 )2 )

where x is the horizontal distance from the side of the doorway and h is the height of the arch at that spot. All measurements in total inches.

Hope this helps,
Stephen La Rocque.

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