ForHosting KIT · Math

Prolate spheroid calculator

Enter the two semi-axes of a prolate spheroid and get its volume, surface area and eccentricity: with a of 5 and c of 3, the volume is 188.495559, the surface 165.793062 and the eccentricity 0.8. Free in your browser.

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Worked example

If you enter

A: 5
C: 3
Decimal places: 6

You get

Volume: 188.495559
Eccentricity: 0.8
Surface area: 165.793062

Volume, eccentricity and surface of prolate spheroid a>c. Search intent: prolate spheroid volume. No network calls or random seeds are required. Failed jobs are not billed.

What a prolate spheroid is

A prolate spheroid is the solid you get by spinning an ellipse around its long axis, which gives a stretched shape like a rugby ball. It is described by two semi-axes: a, measured along the axis of rotation, and c, the radius around the middle. Because the shape is stretched, a is always the larger one. The volume is four thirds of π times a times c squared, the same as the formula for a sphere with one radius replaced by a. The eccentricity, the square root of one minus c squared over a squared, says how stretched it is: close to zero means almost a sphere, close to one means long and thin.

Using the results without mistakes

Enter half-lengths, not full lengths: a is half of the total length and c is half of the width. Both must be in the same unit, and the volume then comes out in that unit cubed and the surface area in that unit squared. Swapping a and c is the most common slip, because it describes a flattened shape instead. If you ask for decimals, each value is rounded once at the end, so with a of 5 and c of 3 you see 188.495559 for the volume and 165.793062 for the surface.

What you can do with it

Teaching

Pinned goldens.

FAQ

What is the formula for the volume of a prolate spheroid?

Multiply four thirds of π by a and by c squared. With a semi-major axis of 5 and a semi-minor axis of 3 that is 60π, about 188.495559 cubic units.

How is the eccentricity of a prolate spheroid calculated?

Divide c squared by a squared, subtract the result from one and take the square root. For a of 5 and c of 3 the eccentricity is 0.8; the closer it gets to one, the more stretched the shape.

Why must a be larger than c in a prolate spheroid?

Because a prolate spheroid is stretched along its axis of rotation, so the semi-axis along that axis, a, is the longer one. If a equalled c you would have a sphere, and a smaller a would describe a flattened shape instead.

Can I choose how many decimals the results show?

Yes. Set the number of decimals and each result is rounded once, at the very end, with halves rounded up. The example values on this page use six decimals.

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

POSThttps://api.kit.forhosting.com/math/prolate-spheroid

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/math/prolate-spheroid \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"a":5,"c":3}'

const res = await fetch("https://api.kit.forhosting.com/math/prolate-spheroid", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "a": 5,
    "c": 3
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/math/prolate-spheroid",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "a": 5,
    "c": 3
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/prolate-spheroid", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"a":5,"c":3}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"a":5,"c":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/prolate-spheroid", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "a": 5,
  "c": 3
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "math.prolate_spheroid",
  "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

Per request$0.002

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

The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos.

How it works

Volume, eccentricity and surface of prolate spheroid a>c. V=(4/3)π a c²; e=√(1−c²/a²). No network calls or random seeds are required. Failed jobs are not billed. No network calls or random seeds are required. Failed jobs are not billed. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos.

Inputs, limits and rounding

Optional decimals use half-up once. Optional output fields are omitted, never null. No network calls or random seeds are required. Failed jobs are not billed. No network calls or random seeds are required. Provide required JSON fields with DOM-safe names. Invalid types raise invalid_input. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks,

When to use it

Use when search intent matches. Volume, eccentricity and surface of prolate spheroid a>c. EN-only delivery; Nitro hydrates locales. No network calls or random seeds are required. Failed jobs are not billed. No network calls or random seeds are required. Failed The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos. The calculator is free in the browser, deterministic, and identical on the API path. Inputs are validated before arithmetic so invalid payloads fail with invalid_input. Half-up rounding is applied once from raw IEEE values when decimals are requested. Use the same JSON from spreadsheets, CMS hooks, automation, or teaching demos.

Integration use cases

Quick check

Free widget.

Automation

API same JSON.

Technical questions

Free?

Yes tier A.

Network?

No.

Bad input?

invalid_input.

API?

Yes.

Locales?

EN-only; Nitro hydrates.

Read the full KIT documentation →

Run — free