Arc / Circle

What it's for

Finding the radius of a curve you can only measure across, and setting it out without reaching the centre of the circle. You walk away with the radius, the sweep and arc length, a stud-by-stud schedule for an arched wall with the top cut angle for each stud, a table of ordinates to mark the curve from the chord, and a printable template or set-out sheet. It is geometry, not a code check.

Step by step

The example is the screen as it opens: an arch 15' 0" across that rises 5' 0" at the middle, framed on the inside with studs at 16".

Arc / Circle inputs at their defaults: KNOWN Chord + Rise (the others are Dia + Angle and Dia + Arc), CHORD (RUN) 15' 0", SEGMENT RISE 5' 0", then ARCHED WALL with SIDE Inside, STUD SPACING 16" picked among 12", 16", 19.2" and 24", BASE HEIGHT 0" with a note that it runs from the top of the bottom plate up to the chord, SET-OUT INCREMENTS 12 and WASTE FACTOR 10%. The answer bar under the title reads R 8' 1-1/2", with Dia 16' 3" · sweep 134.76° under it. — What you know about the curve, then how the arched wall is framed.

KNOWN — which two things you can measure. Chord + Rise is the one you can take off a wall or an opening with a tape. Dia + Angle and Dia + Arc are covered below. Leave it at Chord + Rise.
CHORD (RUN) — the straight line between the two ends of the arc, not the length of the curve. Measure springing to springing. 15' 0" here.
SEGMENT RISE — how high the arc stands above the chord at the middle. Measure square off the chord at its midpoint. 5' 0" here.
Under ARCHED WALL, SIDE — Inside frames the space between the chord and the curve. Outside frames above the curve, up to a level line at the crown.
STUD SPACING — 12", 16", 19.2" or 24" on centre.
BASE HEIGHT — added to every stud, for an arch that sits on top of a straight wall. Inside, it is measured from the top of the bottom plate up to the chord; outside, from the crown up to the underside of the top plate. The note under the field says which. It opens at 0".
SET-OUT INCREMENTS — how many equal stations along the chord you will measure the curve up from. More stations give a smoother curve. 12 here.
WASTE FACTOR — added to the materials you buy by the foot or the sheet. It does not change any dimension. It opens at 10%.
Read RESULTS: R 8' 1-1/2", then dia 16' 3" · sweep 134.76°.
Read the stud schedule under ARCHED WALL — INSIDE: CENTRE 5' 0", then AW1 ≈ 4' 10-13/16" LP @ 1' 4" cut 9.45° off square, and on out to AW5.
Check the drawing: the arch with its studs on the 15' 0" chord, and the stud table beside it.
Set the curve out with the RUNNING MARK-OUT card.
Print the template or the set-out sheet.
Read MATERIALS, then use the buttons at the foot of the screen.

Arc results, scrolled so the last inputs sit above them. The answer bar keeps R 8' 1-1/2" with Dia 16' 3" · sweep 134.76°. RESULTS reads Dia 16' 3" · sweep 134.76°; Arc ≈ 19' 1-5/16" · chord 15' 0" · rise 5' 0"; Segment ≈ 54.1972 sq ft (≈ 7804.40 sq in) · pie ≈ 77.6347 sq ft; and Full circle area ≈ 207.3942 sq ft with circumference ≈ 51' 0-5/8". Under ARCHED WALL — INSIDE come CENTRE 5' 0" and AW1 to AW5, each with a length to the long point (LP), a station and a cut in degrees off square, AW1 ≈ 4' 10-13/16" LP @ 1' 4" cut 9.45° off square. Two notes explain where the studs run and what LP means. ORDINATE SET-OUT reads 12 increments @ 1' 3" — full station/offset table below. — The radius, the circle figures, then one row for each stud.

The other two ways in

Dia + Angle swaps the first two inputs for DIAMETER and ARC ANGLE, the sweep in degrees. Dia + Arc asks for DIAMETER and ARC LENGTH, the distance measured around the curve. Use them when the drawings give you the circle. Everything below the first two inputs works the same way.

Reading the results

The headline is the radius. It also shows in a strip under the title whenever the results card is scrolled off screen. A ≈ in front of any figure means it has been rounded for display.

The circle figures. Diameter and sweep first (the line the strip under the title keeps), then arc length, chord and rise. The area of the segment (between chord and curve) and of the pie slice back to the centre, each in square feet and square inches. The area and circumference of the full circle. A curve's area has π in it, so it is never an exact decimal: each area carries ≈, as in Segment ≈ 54.1972 sq ft (≈ 7804.40 sq in) on the defaults.

Arched wall. CENTRE is the stud on the chord midpoint. AW1 is the first stud out from it, AW2 the next, and so on. Each line gives the length to the long point (LP), the station of the stud's centre measured from the chord midpoint, and the top cut in degrees off square. The long point is the stud face the curve is highest over — on an inside wall the face nearer the crown, on an outside wall the face nearer the springing — and it is where you hook the tape and mark the stud before cutting the bevel; a length at the stud's centre would leave it about 3/4" short. The two notes under the list say where the lengths run: inside, from the top of the bottom plate to the underside of the curved top plate, which is the arc you entered (so nothing is deducted for that plate); outside, from the top of the curved plate up to the underside of the top plate. Each length repeats on both sides, so AW1 is two studs, one each side.

Past a half circle (a horseshoe arch, where the rise is more than half the chord) the curve bulges out past each springing, where no station along the chord reaches. A line under the ordinates says how far it bulges and where the circle's centre is, so you can set that part out from the centre.

Ordinate set-out gives the number of increments and their spacing — 12 increments @ 1' 3" here. The table itself is in the mark-out card.

There is no pass or fail on this screen and nothing turns red.

Arc / Circle scrolled to the drawing. Above it the end of RESULTS and the link Check before you cut · Measure twice. The ARC SET-OUT drawing is a segmental arch on a 15' 0" chord with its studs drawn, the ordinate offsets between them dimensioned, a corner table headed AW# LP OFF SQ (CTR 60" 0.0°, then 1 to 5, the first ≈ 58-13/16" 9.4°), notes reading 12 ORDINATES @ 1' 3" FROM LEFT SPRINGING, OFFSETS SQUARE OFF THE CHORD · RISE 5' 0", ARC ≈ 19' 1-5/16" · 134.76° SWEEP, INSIDE ARCHED WALL and TICK OFFSETS EACH END, and the title strip ARC SET-OUT R 8' 1-1/2" NOT TO SCALE. Below it the RUNNING MARK-OUT card opens, MARK 1 OF 13 with a speaker button and the ORDINATES, FROM CROWN and STUDS tabs. — The drawing, and the top of the mark-out card with its three tapes.

The drawing shows the arch on its chord with every stud drawn, and a table of stud lengths and cut angles in the corner. Pinch to zoom in on the table, or tap the expand button at the top right to open the drawing full screen. See Reading the drawing.

The mark-out card has three tapes:

ORDINATES — hook one end of the chord. At each station, square off the chord by the second figure and mark the arc. The marks are numbered #0, #1 … #12. In the example #1 is 1' 3" along and ≈ 2' 0-13/16" up. Join the marks with a batten. No circle centre needed.
FROM CROWN — the same curve measured down from a level line struck at the crown. Use it when you can reach the top line and not the chord.
STUDS — stud centres, hooked from the chord midpoint, the same both ways.

Tap ✓ MARKED — NEXT after each mark to keep your place. How the card works, and how to have it read the marks aloud, is in Mark-out and speak.

The RUNNING MARK-OUT card, MARK 1 OF 13, on its ORDINATES tab with the speaker button at the right: the whole list #0 0" → 0" through #12 15' 0" → 0", the hook note, the green ✓ MARKED — NEXT button and RESET. Below it the card TEMPLATE — TOO BIG FOR 1:1: true size 15' 0" × 5' 0" needs 1 : 24.4 to fit a page, so no curve is drawn, the sheet prints the ordinate set-out instead, with a PRINT SET-OUT SHEET button. — The whole ordinate tape and the template card.

The template card tells you which sheet you will get before you spend the paper.

TEMPLATE — FITS AT 1:1 — the curve is small enough to print at true size on one page. The button reads PRINT 1:1 TEMPLATE — 100% SCALE, DO NOT FIT TO PAGE. Print at 100% and measure the 1" check square on the sheet before you trace anything.
TEMPLATE — TOO BIG FOR 1:1 — as in the example, where a 15' 0" × 5' 0" arch would need 1 : 24.4. The calculator will not print a shrunken curve, because a curve traced off a reduced print is scrap. The button reads PRINT SET-OUT SHEET and the sheet carries the ordinate table instead.

More on printing true to size is in Cut sheets and 1:1 templates.

Materials. The stud line is a net count — 11 pcs here, five each side and the centre stud — with no waste added, because every stud is a different length and a spare fits no station. The curved plate line depends on the bow. When a straight board can hold the curve — the board width it needs is no more than a 2x12's 11-1/4" — it is the arc length in lineal feet plus your waste, and its name tells you how wide the board has to be. When no board is that wide, it is Curved plate — plywood segments, in 4 × 8 sheets: the plate cut from two layers of 3/4" plywood in segments, joints staggered, with the number of segments, each segment's chord and blank size, and your waste. On the defaults that is 3 segments a layer on 4 sheets.

The buttons. CUT SHEET shares a PDF with the drawing, every figure, the stud schedule and the ordinates, and under INPUTS every value you entered. The stud schedule opens with the same two notes as the card, what the lengths run between and what LP means, and History keeps them with the job. The list icon at the right end of the same button shares the material list alone. SAVE puts the job in History. CUT LIST starts a checkable cut list on the Lock Screen and SEND LIST sends one. The cut list is not the same as the material list: it breaks the stud line out one row per stud length, two of each and one centre stud, so you can tick them off at the saw. CUT SHEET, the list icon in it, CUT LIST and SEND LIST stay dim until you change at least one input. SAVE stays lit, but while every input is still a default a tap only answers Enter your job's numbers first and saves nothing.

Arc / Circle is a Pro calculator, and the cut sheet, the cut list on the Lock Screen and History are part of Pro too. This help is free. See Free and Pro.

Common mistakes

Entering the length around the curve as CHORD (RUN). The chord is the straight line between the ends.
Measuring SEGMENT RISE off the floor. It is measured from the chord. If the arch sits on a straight wall, the wall's height goes in BASE HEIGHT.
Cutting one of each stud. The schedule lists one side of the wall. Every AW length is cut twice.
Measuring a stud at its centre. The lengths are to the long point; mark there, then cut the bevel.
Taking off a plate for the curved top. The arc you enter is the underside of the curved plate on an inside wall, so its studs reach it as listed.
Letting the printer fit the 1:1 template to the page. Print at 100% and check the 1" square.