Lumber & Woodwork
Board Foot Calculator
Find the board feet and cost of hardwood or rough lumber from thickness, width and length.
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Lumber & Woodwork
Arch Calculator
Laying out an arched opening, a curved header or an arched window trim? Enter the span and rise to get the radius to swing and the length of the curve.
What is a arch calculator?
A segmental arch is part of a circle. If you know how wide the opening is and how high the arch rises, you can find the circle’s radius — which is what you need to draw the curve with a trammel or string.
How does it work?
The radius comes from the chord–sagitta relationship: R = (rise² + (span ÷ 2)²) ÷ (2 × rise). The central angle and arc length follow from R. Cutting the arch from solid stock? Price the blank with the Board Foot Calculator.
What information is needed?
- Span — the straight width of the opening
- Rise — the height of the arch above that line
How to use the Arch Calculator
- Measure the span between the points where the arch starts.
- Decide on or measure the rise.
- Select Calculate.
- Mark the centre point at radius − rise below the chord and swing the curve.
The formula
In plain terms, here is the calculation the tool runs:
R = (Rise² + (Span ÷ 2)²) ÷ (2 × Rise)
θ = 2 × asin(Span ÷ 2R)
Arc length = R × θ
- R
- Radius
- θ
- Central angle in radians
Worked example
Example
An arched header over a 36" doorway with an 8" rise.
- (8² + 18²) ÷ (2 × 8) = 388 ÷ 16
- R = 24.25"
- θ = 2 × asin(18 ÷ 24.25) = 95.9°
- Arc = 24.25 × 1.673 = 40.57"
Swing a 24 1/4" radius; the curve is about 40 9/16" long.
Frequently asked questions
What if I want a half-circle?
Set rise to exactly half the span. The radius will equal the rise.
How do I draw it without a big compass?
Use a trammel: a straight stick with a nail at one end and a pencil at the radius distance.
Does this work for arch-top windows?
Yes — measure the glass or frame span and rise and use the radius for trim.
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