Spherical cap surface area
Measure just the curved skin of a dome, the part you clad, coat or use for heat transfer: a cap 2 high on a sphere of radius 5 has a curved area of 20π. Heights above the diameter are rejected. Free in your browser.
- Free
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- Your data stays on your device
Run — free
Worked example
- Radius
- 5
- Height
- 2
- Diameter
- 10
- Curved surface area
- 62.8318530718
A spherical cap is the portion of a sphere cut off by a plane. Architects, students, and manufacturing engineers often need only the curved surface area of that dome—not the volume or the flat circular base—when estimating cladding, coating, or heat-transfer surface. This spherical cap surface area calculator takes the sphere radius and the cap height, applies the closed form A equals two pi R h, and returns a stable rounded area. It rejects a height larger than the sphere diameter so impossible geometry never becomes a silent negative or oversized answer.
What a spherical cap is and why curved area equals 2πRh
Cut a sphere of radius R with a plane; the smaller piece is a spherical cap of height h measured along the axis from the cutting plane to the sphere surface. The base of the cap is a circle of some radius a related to R and h by a squared equals h times two R minus h, but you do not need a when you only want the curved skin of the dome.
Classical surface-of-revolution calculus integrates the sphere generators from the plane to the pole and yields the remarkable identity that the curved surface area depends only on R and h: A equals two pi R h. That formula is independent of the base radius a, so a shallow dome and a deep bowl on the same sphere scale linearly with height alone. When h equals R the cap is a hemisphere and A equals two pi R squared, half the full sphere surface four pi R squared.
When h equals the full diameter two R the formula returns four pi R squared, the entire sphere. Height greater than two R cannot occur for a real sphere of radius R; this calculator treats that case as invalid input rather than extrapolating nonsense. Negative or zero radius and height are likewise rejected because they do not describe a solid geometric cap.
Understanding that A is proportional to both R and h helps you catch unit mistakes: doubling the sphere size at fixed relative height multiplies area by four only if h also scales with R; at fixed absolute h, doubling R only doubles the curved area.
Who needs the curved area of a dome
Classrooms and tutoring platforms use a dedicated spherical cap surface area calculator so learners can check two pi R h without confusing it with the volume formula one third pi h squared times three R minus h, or with the full sphere surface four pi R squared. Exam generators need a deterministic oracle that rejects height above the diameter and never invents a base area when the prompt asked only for the curved dome. Failed validation does not consume balance.
Pair this tool with unit converters when your drawings mix millimetres and inches, and document which length unit you fed in so downstream cost models stay honest.
What you can do with it
Classroom and homework checks
Let students submit sphere radius and cap height and verify that curved surface area equals 2πRh before they mix it up with volume or full-sphere area.
Exam and LMS auto-grading
Mark spherical-cap surface items with a deterministic oracle that rejects height greater than the diameter and returns a stable rounded 2πRh.
Dome cladding and coating estimates
Estimate curved dome skin area from known sphere radius and rise in architecture or tank scripts without a full CAD solid-geometry stack.
FAQ
What happens if the cap height is larger than the sphere diameter?
A spherical cap cannot be taller than 2×radius; invalid requests are not billed.
Does this include the flat circular base of the cap?
No. Curved surface area is the dome skin only (2πRh).
What is the difference between this and a full spherical cap calculator?
This tool returns only curved surface area from R and h.
What is the curved area of a cap with radius 5 and height 2?
It is 20π, about 62.831853071796 square units, in the same length unit you used for the radius and the height. The flat circular base is not part of that figure.
Why does the curved area depend only on the radius and the height?
Integrating over the dome gives 2πRh, with no base radius in it. So on one sphere the area grows in step with the height, and at a fixed height doubling the radius only doubles the area.
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/geom/spherical-cap-surface \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5,"height":2}'const res = await fetch("https://api.kit.forhosting.com/geom/spherical-cap-surface", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 5,
"height": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/spherical-cap-surface",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 5,
"height": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/spherical-cap-surface", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":5,"height":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":5,"height":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/spherical-cap-surface", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius": 5,
"height": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.spherical_cap_surface",
"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.
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
Input contract, formula, and response shape
If height is strictly greater than two times radius the task fails with a clear message that a spherical cap taller than the diameter is not defined. Height equal to the diameter is allowed and returns the full sphere surface. Example: radius five and height two yields curved surface area twenty pi, about sixty-two point eight three one eight five three zero seven one eight square units in the same length unit you used for R and h. POST a JSON body with two required positive finite numbers: radius (aliases r, R) for the parent sphere, and height (aliases h, H, cap_height) for the axial cap depth. Each field may arrive as a JSON number or as a numeric string with optional sign, decimal, or scientific exponent. Missing fields, empty strings, non-finite values, zero, and negative values fail with invalid_input naming the field; those failures are not billed. On success the body echoes radius, height, and diameter (two times radius), returns curved_surface_area equal to two pi times radius times height rounded to twelve decimal places for stable goldens across runtimes, and includes the formula string A equals two pi R h for display beside the answer. The API is unit-agnostic: it multiplies the numbers you send and assumes radius and height share one consistent length unit; attach your own labels (square metres, square inches) in the client. This endpoint does not compute cap volume, base area, or base radius; use a full spherical-cap metrics tool when your workflow needs the whole suite from the same two inputs.
Where a spherical cap surface area API earns its keep
Architecture, tank design, and cladding estimates often know sphere radius and dome rise from drawings and need only the exterior skin area for material or coating quotes; calling a tiny parse API avoids pulling a full solid-geometry package for one product. Keep requests small, pin the rounded curved_surface_area in tests, and you get reproducible areas for homework, LMS plugins, CAD helpers, and edge workers without standing up symbolic math infrastructure of your own.
Technical questions
How do I calculate the curved surface area of a spherical cap?
Send positive finite radius and height. The API returns curved_surface_area = 2 × π × R × h, rounded to twelve decimal places for stable results.
What does the spherical cap surface area calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.