Lesson 10 of 13
Circles and Arcs
Sooner or later somebody wants a curve. An arched opening, a radiused walk, a bullnose step. The problem is almost never the curve itself — it is that the center of the circle is out in the parking lot and you cannot swing a string to it.
Step one
Read the material
Three ideas. The second one is the formula worth writing inside your toolbox lid.
Concept one
Halve the diameter first, every single time
Circumference is pi times the diameter. Area is pi times the radius squared. Those two sentences use different numbers, and that is where the mistakes live.
Nearly every circle on a job site is described by its diameter, because that is what a tape reads across the thing. An eight foot round patio, a four inch pipe, a twelve inch sonotube. The formula for area wants the radius. So halve it first, write the radius down, and then square it.
Get that backwards and your answer is four times too big, which is a mistake large enough that you will notice it at the concrete plant rather than at the desk.
Concept two
Radius from a chord and a rise
Here is the practical problem. Somebody hands you an arched opening and says make the trim follow it. You can measure straight across the opening, and you can measure how high the arch bellies up in the middle. You cannot reach the center of that circle, because it is somewhere behind the wall.
Those two measurements are enough. The distance across is the chord. The height at the middle is the rise. The radius is the chord squared, divided by eight times the rise, plus half the rise.
An opening 48 inches across with an 8 inch rise: 48 squared is 2304, eight times 8 is 64, and 2304 over 64 is 36. Add half the rise, which is 4, and the radius is 40 inches. Now you can strike that arc from a point on the floor, or build a trammel to it, without ever finding the true center in the wall.
Concept three
The arc is longer than the chord, and that is what you buy
The material follows the curve, not the straight line under it. Order to the chord and you will be short every time.
Arc length is the radius times the included angle measured in radians. In practice: find the radius, find the angle the arc sweeps, then take that fraction of the full circumference. The 48 inch opening above sweeps about 73.7 degrees, roughly a fifth of a circle, and its arc measures 51 and a half inches against a 48 inch chord. Three and a half inches you would not have bought.
On a gentle curve the difference is small. On a tight one it is not. Always price and cut to the arc.
Step two
Work it with your hands on it
Set the chord and the rise, the two things you can actually measure. Everything else is computed from them and drawn to scale.
Hands on it, off the screen. Find an arch, a curved counter, a rounded step. Measure straight across it and measure the height at the middle. Work the radius before you look at this panel, then check yourself.
Then drive a nail at that radius from the middle of the chord, hook a tape on it, and swing the curve on a sheet of scrap. If your line lies down on the real arch, you own the method.
Step three
Prove you understood it
One question per idea, new numbers every time.
All three solid.
You can get an area without doubling it, find a radius you cannot reach, and buy to the arc instead of the chord. Lesson eleven is offsets, which is the same triangle again under two more names.
Trades Geometry
Lesson ten of thirteen. Written by our founder, Dr. Gene A Constant, and donated to the Foundation. Published by Global Sovereign University Press.
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