Arc Chord and Mid-Ordinate Check Calculator

SettingsSettings for this calculationUS

Market

Imperial · sales tax

Design Radius

The radius the arc is supposed to have been struck at.

For a basketball three-point line this is measured from the point on the floor directly beneath the centre of the ring, not from the backboard face and not from the midpoint of the baseline. Establish that point from the goal as installed, because goal installation carries its own tolerance and rarely lands exactly where the drawing put it.

Spacing Between Check Stations

The distance along the arc from one check point to the next.

Measured around the curve rather than straight across, because that is how a tape follows a struck line. Close spacing catches a local wobble; wide spacing catches a wrong radius. Three or four stations spread across the arc will find both.

Length of Arc Being Checked

How much of the curve you are checking, measured around it.

This sets how many stations fall on the arc and what angle it turns through. Where a line runs into a straight tangent, check to the tangent point and no further — the geometry beyond it is a straight line and will fail an arc check for the right reason.

Mid-ordinate at each station

0.8348 in

High confidence

A chord reading short against these figures means the pivot moved or the trammel shortened; a mid-ordinate reading high means the radius struck was tighter than the design. The two together tell you which.

Chord between adjacent stations: 3.5 ft
Check stations that fall on the arc: 11 stations
Angle the arc turns through: 101.57 degrees
Chord across the whole arc: 34.09 ft

Then change the inputs to see how far the answer moves.

Show calculation logic

How this was calculated

Formula source(s)

Circular curve setting out: for an arc distance s on a radius R the half angle is s ÷ 2R, the chord across it is 2R·sin(s ÷ 2R), and the mid-ordinate from that chord to the curve is R·(1 − cos(s ÷ 2R)). Geometry throughout; no coefficient is involved.
FIBA Official Basketball Rules — the three-point arc is struck at 6.75 m from the point on the floor vertically beneath the centre of the ring, reducing to 6.60 m where the line runs parallel to the sideline, which is the default radius offered here

Inputs used

Design Radius: 22 ft
Spacing Between Check Stations: 3.5 ft
Length of Arc Being Checked: 39 ft

Intermediate steps

Chord between adjacent stations: 3.5 ft
Check stations that fall on the arc: 11 stations
Angle the arc turns through: 101.57 degrees
Chord across the whole arc: 34.09 ft

Final result0.83 in

Confidence note: A chord reading short against these figures means the pivot moved or the trammel shortened; a mid-ordinate reading high means the radius struck was tighter than the design. The two together tell you which.

What this calculation does not cover

Arcs longer than a semicircle are reported at the clamp, where the chord equals the diameter — check such a curve in halves instead.
This proves the curve against its own radius. Whether the pivot was in the right place is a separate check, made from the goal or the datum rather than from the arc.

Add the equipment this sizes

This result is a specification — 0.8348 in — not a quantity. Put the thing it sizes into your project: how many, what you call it, and your supplier’s price.

QtyItemPrice each ($)

Schematic, drawn to the proportions you entered — not to scale on screen.

Computed in your browser — nothing you enter is uploaded. Presented in US customary units and US trade terminology. Where a formula follows a published standard, that standard and its edition are cited beside it on this page; where none governs, the page says so. Local amendments override model codes — verify against the code in force where you build.

Sources checked 2026-09-02 · in the site-wide review of 2026-09-06 · v1.0.1

Regulatory standards & verification citations2

Circular curve setting out: for an arc distance s on a radius R the half angle is s ÷ 2R, the chord across it is 2R·sin(s ÷ 2R), and the mid-ordinate from that chord to the curve is R·(1 − cos(s ÷ 2R)). Geometry throughout; no coefficient is involved.
FIBA Official Basketball Rules — the three-point arc is struck at 6.75 m from the point on the floor vertically beneath the centre of the ring, reducing to 6.60 m where the line runs parallel to the sideline, which is the default radius offered here

Cite this page

Your workspace

Most jobs need more than one number. Add the calculators you need next and they open right here, underneath this one — your figures stay on screen and nothing is lost to a page change.

Now that you have the number

These guides cover the work this quantity is for — the first ones run this calculator inside the section that raises the question.

Setting Out a Building With Profiles and a Tapeuses this calculator

Pegs, string and one tape the day before the digger: finding the building line, proving the base by its diagonals, and landing the curve where the drawing put it.

Turning a Brick Arch Over an Openinguses this calculator

One number — the rise — settles what family the arch belongs to, how hard it pushes sideways, and the taper on every wedge you cut.

Building a Curved Drywall Wall

How to calculate arc chord and mid-ordinate check in 4 steps

Design RadiusThe radius the arc is supposed to have been struck at.

Spacing Between Check StationsThe distance along the arc from one check point to the next.

Length of Arc Being CheckedHow much of the curve you are checking, measured around it.

Mid-ordinate at each stationThe tool computes the mid-ordinate at each station from those figures and shows the formula, its sources, and a confidence rating alongside it.

Design Radius	Mid-ordinate at each station (in)
10 ft	1.61
20 ft	0.807
30 ft	0.538
40 ft	0.404

Design Radius

Mid-ordinate at each station (in)

10 ft

1.61

20 ft

0.807

30 ft

0.538

40 ft

0.404

Frequently asked questions

Why check the mid-ordinate rather than just the chord?

Because the offset is the sensitive one. Between two stations a metre apart on a large radius the chord is barely shorter than the arc, so a radius error hides in the third decimal place — while the offset from that chord to the curve changes proportionally and is measured in millimetres you can actually read. The chord tells you the stations are where you thought; the offset tells you the curve between them is right.

Why strike an arc with a trammel instead of a string?

Nylon stretches under the pull needed to keep it straight, and how hard someone pulls varies with how hard they are concentrating. Over a radius near seven metres that is easily ten millimetres of wander around the curve, all of it invisible until the line is painted. A rigid bar with a pin at one end and a scribe at the other gives the same radius every time.

How many stations should I check?

Three or four across the arc is enough to separate the two failures that matter. Stations close together find a local wobble where the trammel jumped or the surface caught it; stations far apart find a radius that was wrong from the start. Repeat the readings after masking as well as after striking, because tape wanders when it is pulled around a curve.

What does a chord reading short actually mean?

Usually that the pivot shifted between one part of the strike and another, which shows up as a chord that no longer matches its neighbours rather than as one uniformly wrong figure. A radius struck tight from a stationary pivot gives short chords everywhere and a mid-ordinate reading high, so comparing the pattern across stations tells you which of the two happened.