Arches, tanks and lenses
A circular segment is the region of a circle cut off by a straight line, the chord. You meet it in arched windows and doorways, in the cross-section of liquid in a horizontal tank, in road curves and in the profile of a lens. Two measurements usually describe it: the chord length (width) and the height, also called the sagitta or rise.
Finding the radius from width and height
When you measure an existing arch, you rarely know its radius. Use:
r = c² ÷ (8h) + h ÷ 2
where c is the chord and h the height. This comes from the right triangle formed by the radius, half the chord and the distance from the center to the chord.
The other formulas
With the radius r and the height h:
- Half the central angle: α = arccos((r − h) ÷ r)
- Chord: c = 2 × √(h × (2r − h))
- Arc length: s = r × 2α (α in radians)
- Area: A = r² × α − (r − h) × √(2rh − h²)
When the height is more than the radius, the segment is larger than half the circle, and the formulas still hold.
Example: an arched window
A window arch is 48 inches wide and rises 12 inches:
- Radius: 48² ÷ (8 × 12) + 12 ÷ 2 = 24 + 6 = 30 in
- Central angle: 2 × arccos(18 ÷ 30) = 106.3°
- Arc length: 30 × 1.855 ≈ 55.6 in, useful for bending trim or ordering a curved head
- Area: 900 × 0.9273 − 18 × 24 ≈ 402.6 sq in, about 2.8 square feet of glass
Example: liquid in a horizontal tank
A round tank with a 30-inch radius holds liquid 12 inches deep. The liquid’s cross-section is the same segment as above, 402.6 sq in. If the tank is 72 inches long, the liquid volume is 402.6 × 72 ≈ 28,990 cubic inches, about 125 US gallons.
Tips
- Use the same unit for every length; the area comes out in that unit squared.
- To lay out an arch on site, a string fixed at the center with the computed radius draws the curve.
- For a very flat arch, measure the height carefully: a small error changes the radius a lot.