How to Find a Radius from a Chord and a Rise

Why this matters

Curves show up in work that is otherwise all straight lines: a curved planting bed the customer wants extended, an arched opening that needs trim or a new head jamb, a curved wall that a floor has to be scribed to, a radius patio edge, a pool coping replacement. You almost never get told the radius. You can see the curve, but its center is somewhere off in the yard or inside a wall where you cannot put a tape on it.

You do not need the center to find the radius. Two measurements taken entirely on your side of the curve give it to you: the chord, a straight line stretched between two points on the curve, and the rise (also called the sagitta), the distance from the middle of that line out to the curve. From those, one formula gives the radius, and with the radius you can reproduce, extend or match the curve.

The geometry, labeled

              M  <- curve at the midpoint
         . '  |  ' .
      .'      | h    '.      h = rise = 1 ft
   A +--------+--------+ B   chord A-B = c = 8 ft
              |              half chord = 4 ft
              |
              | R - h = 7.5 ft
              |
              O  <- center of the circle
   R = O to A = O to B = O to M = 8.5 ft

The line from the center O through the midpoint of the chord is square to the chord and passes through the highest point of the curve, M. That is why the rise has to be measured at the exact middle of the chord and square to it: it is the only place the rise lines up with the center.

Where the formula comes from

Look at the right triangle made by the center O, the chord midpoint, and the end of the chord A. Its legs are the half chord (c / 2) and the distance from the center to the chord (R - h). Its long side, from O to A, is the radius R. So by the Pythagorean theorem:

(R - h) squared + (c / 2) squared = R squared

Multiply out the first term: R squared - 2Rh + h squared + c squared / 4 = R squared. The R squared on each side cancels, leaving 2Rh = h squared + c squared / 4. Divide both sides by 2h:

R = c squared / (8 x h) + h / 2

Both terms come out in the same unit as c and h, as long as c and h are in the same unit. That condition is the whole trap, and it gets its own step below.

Step 1: Pick the chord ends as far apart as the curve allows

Choose two points on the curve and mark them. For a full arched opening, the chord is usually the springing line, where the arch meets the straight jambs. For a bed edge or a curved wall, take the longest stretch that is clearly one continuous curve.

Why long: the rise grows with the square of the chord. Double the chord on the same curve and the rise becomes about four times as large, so the same measuring error is a much smaller share of it. The sensitivity section below shows how much this matters on a shallow curve.

What breaks if you skip it: a short chord on a gentle curve gives a rise of an inch or two, and a quarter inch misread there moves the radius by feet.

Step 2: Stretch a line between the ends and measure the chord

Pull a mason's line or a taut string between the two marks, or snap a chalk line on a floor, or hold a straightedge across an opening. Measure the chord from mark to mark along the line.

For a curve on the ground, set a stake at each end and tie the line tight to them. Stakes that will be driven only a few inches for a layout line are not excavation, but if the layout leads to digging (a bed edge trench, a footing for a curved wall), the utility locate request comes before the shovel.

Step 3: Measure the rise at the exact middle, square to the chord

Find the midpoint of the chord by measuring half the chord from one end. From that mark, measure straight out to the curve, square to the line. A framing square against the string or a second tape held at a right angle keeps it square.

Why the middle and why square: a rise measured off-center or on a slant is longer or shorter than the true rise and does not line up with the center. The formula has no way to know, and returns a confident wrong radius.

Step 4: Put both numbers in the same unit, then compute

Convert so that c and h are both in feet or both in inches. Then:

Worked example, a curved bed edge. The chord is 8 ft and the rise is 1 ft.

• c squared = 8 ft x 8 ft = 64 sq ft
• 8 x h = 8 x 1 ft = 8 ft
• c squared / (8h) = 64 sq ft / 8 ft = 8 ft
• h / 2 = 1 ft / 2 = 0.5 ft
• R = 8 ft + 0.5 ft = 8.5 ft

The same numbers in inches: c = 96 in, h = 12 in. 96 x 96 = 9,216 sq in; 8 x 12 = 96 in; 9,216 / 96 = 96 in; plus 12 / 2 = 6 in; R = 102 in. 102 in / 12 = 8.5 ft. Same answer.

The mixed-unit slip. With the chord in feet (8) and the rise in inches (12): 64 / 96 = 0.67, plus 6, gives "6.67". Not feet, not inches, not anything. A sanity check catches it: the radius of any arc is at least half its chord (exactly half for a half circle), so a radius smaller than 4 ft for an 8 ft chord is impossible.

Step 5: Check the radius back through the triangle

Put the answer back into the Pythagorean check: (R - h) squared + (c / 2) squared should equal R squared.

• R - h = 8.5 ft - 1 ft = 7.5 ft; 7.5 x 7.5 = 56.25 sq ft
• c / 2 = 4 ft; 4 x 4 = 16 sq ft
• 56.25 + 16 = 72.25 sq ft
• R squared = 8.5 x 8.5 = 72.25 sq ft

They match, so the arithmetic is right. This check does not test your measurements, only your math. The next step tests the measurements.

Step 6: Prove the curve is a true circle before you trust it

A curve you can see is not necessarily part of a circle. Elliptical arches, arches built from two or three radii, and a bed edge someone laid out by eye all have a chord and a rise, and the formula will happily give a radius for each. To test, predict the offset from the chord at a second point and measure it.

At a point x out from the chord's midpoint, the offset from the chord to a circular curve is the square root of (R squared - x squared), minus (R - h).

For the bed edge, take the quarter point, 2 ft out from the midpoint (halfway to the end of the chord):

• R squared - x squared = 72.25 - 4 = 68.25 sq ft
• square root of 68.25 = 8.261 ft
• offset = 8.261 ft - 7.5 ft = 0.761 ft
• 0.761 ft x 12 = 9.14 in, about 9-1/8 in

Measure square off the string 2 ft from the midpoint on each side. If both read about 9-1/8 in, the edge is a circular arc with an 8.5 ft radius. If they read noticeably more or less, or the two sides disagree, the curve is not one circle; split it into shorter sections and give each its own chord, rise and radius.

Step 7: Use the radius to reproduce or extend the curve

The center sits on the line through the chord midpoint and the top of the curve, on the side away from the bulge, at R - h from the chord (7.5 ft in the example). Measure 7.5 ft from the chord midpoint along that line, square to the chord, and set a pin. Tie a non-stretch line to it, mark 8.5 ft, and swing the arc. To extend the bed, keep swinging past the old ends.

For length along the curve (edging, flexible trim, kerfed base), the arc is longer than its chord. The angle the arc spans is 2 x the arcsine of (c / 2R): c / 2R = 4 / 8.5 = 0.4706, its arcsine is 28.07 degrees, so the arc spans 56.15 degrees, which is 0.980 radians. Arc length = R x angle in radians = 8.5 ft x 0.980 = 8.33 ft. A shortcut that is good for shallow arcs gives the same answer here: arc length is about c + 8h squared / (3c) = 8 + 8 / 24 = 8.33 ft. Order the edging from 8.33 ft, then add the shop's waste factor.

How sensitive the answer is to the rise

The rise is the measurement that drives the error, and how much depends on how shallow the curve is.

The bed edge again, rise misread by 1/4 in. With a rise of 11-3/4 in (0.979 ft) the radius comes out 8.66 ft; with 12-1/4 in (1.021 ft) it comes out 8.35 ft. A quarter inch either way moves the radius by about 2 in, which is about 2 percent. Fine for a bed, marginal for a curved head jamb.

A shallow curve, same chord. Say a curved wall gives an 8 ft chord with a rise of only 3 in (0.25 ft). R = 64 / 2 + 0.125 = 32.13 ft. Misread the rise by 1/4 in: at 2-3/4 in the radius is 35.02 ft, at 3-1/4 in it is 29.67 ft. The same quarter inch now moves the radius by 2.5 to 2.9 ft, 8 to 9 percent, because a quarter inch is a twelfth of a 3 in rise.

The fix is a longer chord. On that same 32.13 ft radius, a 16 ft chord has a rise of about 12.1 in, and a quarter inch is back down to about a 2 percent error. As a rough rule for shallow curves, the radius error in percent is about the same as the rise error in percent, so make the rise as large as the curve allows.

How to know you got it right

• Chord and rise were in the same unit, and the radius is more than half the chord.
• The Pythagorean check closes.
• The predicted offset at the quarter points matches what you measure on both sides.
• When you swing the radius from the pin you set, the arc passes through both chord ends and the top of the curve.

References

• See related: Common Unit Conversions + Math Formulas
• See related: Circles in the Field: Area, Circumference and the Diameter Trap
• See related: How to Square a Layout with Diagonals and a Triple
• See related: The Measure and Takeoff Basics