Five numbers describe the whole curve
A circular curve is fixed by its radius and the angle it turns through. Everything else follows: the tangent distance from the corner back to where the curve starts, the arc length along it, the long chord end to end, the middle ordinate from the chord out to the curve, and the external distance from the corner in to the curve. Give any two of the right pair and the rest fall out.
Which of them you want depends on how you are laying it out. If you can swing a tape from a centre point, you want the radius and nothing else. If the centre is inside a building, under a pile of gravel or across a road, you cannot, and the curve gets built from a straight line you can actually stretch.
Offsets from a chord
Stretch a string between the two ends of the curve, mark along it at whatever interval you want stakes, and square out from each mark by a calculated offset. The offsets are symmetrical about the midpoint, largest at the middle where they equal the middle ordinate, and zero at the ends. On a 60 foot radius turning 45 degrees, the chord is 45.92 feet and the middle ordinate is 4.567 feet, so the deepest offset is a little over four and a half feet and the marks 5 feet either side of the middle come out at 4.359.
Offsets from the tangent work the same way but from a different straight line: measure along the tangent from where the curve starts and square in. Those offsets start at zero and grow roughly as the square of the distance, so they are tiny near the start and awkward further along. For a short arc off a long straight the tangent method is quicker; for anything that turns much, the chord is easier to hold square.
Getting a radius from something already built
| Chord you stretch | Middle ordinate you would measure | Radius that implies | Radius if the ordinate reads 1/16 in high |
|---|---|---|---|
| 20 ft | 0.250 ft | 200.0 ft | 195.9 ft |
| 40 ft | 1.003 ft | 200.0 ft | 199.0 ft |
| 60 ft | 2.263 ft | 200.0 ft | 199.5 ft |
All three rows are the same true 200 foot curve measured three ways, and all three are measurable with a tape and a string. What changes is how much a small error costs. On the 20 foot chord a sixteenth of an inch of error in the ordinate swings the answer by about four feet; on the 40 foot chord it is a foot; on the 60 foot chord it is half a foot. The rise sits in the denominator, so a small rise means a small number dividing into a big one, and the result is that flat curves measured over short chords give radii you should not trust to three figures. Stretch the longest chord the site allows.
What is not in here
This is a plane circular curve of constant radius laid out horizontally. It has nothing to say about spiral transitions, about superelevation, about vertical curves, or about how a kerb radius relates to the centreline radius on the same corner — three different lines on one intersection have three different radii and confusing them is the most common way a curve gets staked in the wrong place. If all you need is arc, sector and area for a circle, the circle calculator is the shorter route, and the driveway apron calculator handles whether a vehicle can actually make the turn the radius allows.
Questions people ask
What is the middle ordinate?
The perpendicular distance from the midpoint of the long chord out to the curve. It is the deepest the curve gets away from the straight line between its two ends, and it is the single easiest thing to measure on a curve that already exists: stretch a string across, find the middle, measure out. It is also the largest of the chord offsets, so if you are staking a curve from a chord it is the number to set first and check before doing any of the others.
How do I find the radius of a curve that is already built?
Stretch a chord across it, measure the perpendicular rise at the middle, and the radius comes from those two numbers. The exact form is the chord squared over four plus the ordinate squared, all over twice the ordinate; for flat curves the chord squared over eight times the ordinate is close enough. The catch is sensitivity: on a 40 foot chord with a 1 foot rise, a sixteenth of an inch of measurement error moves the radius by about a foot. Use the longest chord you can stretch, because doubling the chord quadruples the rise and the error matters four times less.
Chord offsets or tangent offsets?
Chord offsets are easier for most of what a curve does on a site. They are symmetrical about the midpoint, the largest one is at the middle and is easy to check, and they stay a manageable size. Tangent offsets grow as roughly the square of the distance along the tangent, so they are near zero at the start of the curve and get large fast, which makes the far end awkward to hold square. Tangent offsets are handy for a short arc coming off a long straight you have already strung.
Which radius do the plans mean?
That is worth settling before anything is staked, because a corner has several. A kerb radius, an edge of pavement radius, a centreline radius and a property corner radius on the same intersection are all different numbers, and each one is offset from the next by the width between them. A curve staked from the right radius on the wrong line is out by that offset the whole way round. Read what the dimension is measured to on the drawing rather than assuming.
Does this handle a spiral or a compound curve?
No. Everything here is a single circular arc of constant radius. A spiral transition changes radius continuously along its length and needs its own geometry; a compound curve is two or more arcs of different radii meeting tangentially and has to be laid out one arc at a time with the tangent point between them established first. If your drawing shows a spiral length or two radii on the same bend, this page will describe a curve that is not the one on the plan.