Circular arch calculator
An arch spanning 8 on a radius of 5 rises 2 at the crown, with a soffit arc of about 9.273 and a chord of 8, in whatever unit you use. A span wider than twice the radius is refused. Free in your browser.
- Free
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- Your data stays on your device
Run — free
Worked example
- Span
- 8
- Radius
- 5
- Rise
- 2
- Arc length
- 9.27295218
- Chord
- 8
- Included angle deg
- 106.2602047083
A circular arch is a constant-radius opening defined by its span (the chord between springing points) and the radius of the circle that carries the soffit. Masons, architects, bridge draughtsmen, and carpentry shops need the rise (sagitta), the arc length along the soffit, and the chord before they cut voussoirs, size a formwork, or check a masonry template. This circular-arch calculator takes those two lengths, rejects a span that exceeds twice the radius, and returns the three geometry values in the same unit you sent. A span equal to twice the radius is a semicircle: the rise equals the radius and the soffit is a half-circumference.
How rise, arc length and chord follow from span and radius
A circular arch is the minor, or at most semicircular, arc of a circle that springs from two points a known distance apart. That distance is the span, and it is also the chord of the arc. The circle's radius is the second input.
From those two lengths every other dimension of a segmental or semicircular arch follows. Let L be the span and R the radius. The half-span is L divided by two.
When L equals 2R the figure is a semicircle: the rise equals R and the soffit is a half-circumference. For a smaller span the rise, also called the sagitta, is the distance from the chord up to the crown: rise h equals R minus the square root of R squared minus (L/2) squared. Equivalently, and more stable when the arch is almost flat, h equals (L/2) squared divided by R plus the same square root.
The included central angle in radians is two times the arcsine of (L/2) over R, or two times the two-argument arctangent of the half-span over (R minus h). The arc length along the soffit is R times that included angle. All lengths share the unit of L and R: metres, millimetres, inches, or feet.
The calculator never converts units; it only computes ratios.
What you send, which units to use, and how validation fails
Send two required numeric fields named span and radius. Both must be finite positive lengths in the same unit. A worked case: span 8 and radius 5 yields rise 2, chord 8, included angle about 106.26 degrees, and arc length about 9.273, because the half-span 4 and radius 5 form a 3-4-5 right triangle from the centre to a springing.
A semicircle with span 10 and radius 5 yields rise 5 and arc length 5π. A span of 12 with radius 5 fails because 12 exceeds twice 5. Failed tasks are not billed.
Upper limits are generous finite lengths; the hard geometric limit is the diameter. The same input always produces the same rounded output, which is the contract estimating sheets, CNC templates, and edge workers rely on when they pin an eight-by-five arch as a two-unit rise.
Where a circular-arch calculator earns its keep
Masonry and concrete-arch shops size voussoirs and formwork from rise and soffit length, not from a sketch that looks about right. A garden-wall or window arch with a known opening and a chosen radius must clear a lintel height; the rise tells you how much headroom the crown steals from the storey, and the arc length tells you how much extrados brick or timber you need along the curve.
Timber-frame and metal-fabricator shops that roll or kerf a circular head cut the chord first, then need the rise to set the jig and the arc length to buy the right stock. Curriculum authors and LMS auto-graders pin goldens: span 8 and radius 5 must stay rise 2. The tool is a circular constant-radius arch, not a catenary, ellipse, three-centred basket handle, or Gothic pointed pair of arcs; those are different search intents.
It also does not size loads, mortar joints, or abutment thrust. If you later need a sector area from a central angle, a chord from an angle, or a stair rise and run, treat those as the geometry and stair pages they already are rather than overloading this arch landing.
What you can do with it
Segmental bridge and culvert soffits
A flat circular arch is span plus a large radius; the calculator returns the small sagitta and the true soffit length instead of a mistaken semicircle.
FAQ
How do I find the rise of a circular arch from span and radius?
Send finite positive lengths span and radius in the same unit. Span must not exceed twice the radius.
What happens if the span is greater than twice the radius?
A circular arch cannot span more than the diameter of its circle; that geometry has no real rise and is not billed.
Does the calculator handle elliptical or Gothic arches?
No. It is a constant-radius circular arch only. Catenary, elliptical, three-centred, and pointed Gothic openings are different figures and different search intents.
How long is the soffit of an arch with span 8 and radius 5?
About 9.273 in the same unit. The arch covers a central angle of about 106.26 degrees, and the arc length is the radius times that angle in radians; the rise is 2 and the chord stays 8.
What happens when the span is exactly twice the radius?
You get a semicircle: the rise equals the radius and the soffit is half the circumference. A span of 10 on a radius of 5 rises 5 and has an arc length of 5π.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/build/arch \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"span":8,"radius":5}'const res = await fetch("https://api.kit.forhosting.com/build/arch", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"span": 8,
"radius": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/build/arch",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"span": 8,
"radius": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/build/arch", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"span":8,"radius":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"span":8,"radius":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/build/arch", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"span": 8,
"radius": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "build.arch",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
min_length | 1e-15 |
max_length | 1000000000000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
How rise, arc length and chord follow from span and radius
Geometry forbids a chord longer than the diameter, so L must not exceed 2R; a span that exceeds twice the radius cannot sit on that circle and the calculator rejects it with invalid_input. The chord is identical to the span you sent; it is returned as its own field so templates and API clients can print rise, arc_length, and chord as a triple without copying the input. Results are rounded to twelve decimal places so goldens, widgets, and API clients never disagree on a two-metre rise from an eight-by-five opening.
What you send, which units to use, and how validation fails
Aliases opening, clear_span, and L are accepted for span, and r, R, and arch_radius are accepted for radius, so a form that already says opening does not need a rename. Strings that parse as finite numbers are accepted the same as JSON numbers. On success the response echoes the cleaned span and radius, publishes rise as the sagitta from springing line to crown, arc_length as the soffit length, chord as the opening (equal to span), included_angle_deg and included_angle_rad as the central angle subtended by the arch, and formula as the identity that produced them. Zero, negative, missing, or non-finite values fail with invalid_input naming the field so a form can highlight that control. Successful API calls use the published base of $0.002 per item.
Where a circular-arch calculator earns its keep
Bridge and culvert draughtsmen still set segmental arches by span and radius: a flat arch with a large radius and a small rise is a different figure from a semicircle of the same opening, and confusing those two is how a crown hits a soffit that was drawn as a half-circle.
Integration use cases
Masonry window and garden-wall arches
Send the clear opening as span and the chosen circle as radius; use rise for crown clearance and arc_length for the soffit brick or timber run.
Rolled or kerfed circular heads in the shop
Cut the chord first, then set the bending jig to rise and buy stock to arc_length so a timber or metal circular head matches the opening.
Technical questions
Is a semicircular arch allowed?
Yes. When span equals twice the radius the figure is a semicircle: rise equals the radius, the included angle is 180 degrees, and arc_length is π times the radius.
What does the circular arch calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.