Sphere radius calculator
Recover the radius of a sphere from its volume, surface area or diameter: a volume of thirty-six π, about 113.097335529233, gives a radius of 3 and a diameter of 6. Pick the measure you know and enter it. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Value
- 113.0973355292
- From
- Volume
- Value
- 113.0973355292
- Radius
- 3
- Diameter
- 6
The radius of a sphere is the single length that fixes every other classical measure of the ball: the diameter, the surface area, and the enclosed volume. Students, CAD users, inspectors, and process engineers often already know one of those measures and need the radius back — a tank volume from a process sheet, a painted or plated area from a coating quote, or a diameter read from calipers. This calculator inverts the three textbook identities. From volume it uses r equals the cube root of three V over four pi. From surface area it uses r equals the square root of S over four pi. From diameter it uses the half-length r equals d over two.
The three inversions: volume, surface area, and diameter
A sphere is completely determined by one positive radius r. The three measures this tool inverts are the usual school identities written in the same length unit. Volume is V equals four thirds pi r cubed, so solving for the radius is a cube root: r equals the cube root of three V divided by four pi.
Surface area is S equals four pi r squared, so the inversion is a square root: r equals the square root of S divided by four pi. Diameter is the trivial linear relation d equals two r, so r equals d over two. Units stay consistent: if volume is in cubic centimetres the radius comes back in centimetres; if surface area is in square metres the radius is in metres; a diameter in inches yields a radius in inches.
There is no unit conversion here — only the inversion — so mix SI and imperial at your own risk. Because the cube-root and square-root maps are strictly increasing on the positive reals, each valid measurement produces exactly one positive radius.
Where recovering a sphere radius earns its keep
Classrooms and tutoring platforms use a radius-of-sphere calculator so learners can check the cube-root and square-root inversions without a scientific calculator that hides intermediate rounding. Exam generators need a deterministic oracle that marks zero and negative measurements as invalid instead of awarding credit for r equals zero. Manufacturing and inspection notebooks often know a tank or vessel volume, a coated surface area, or a measured diameter and need the matching radius for a drawing note or a CNC stock size.
Failed validation does not consume balance. Keep requests small, pin the rounded radius in tests, and you get a reproducible length for homework, LMS plugins, and edge workers without standing up a computer-algebra system for three textbook identities.
What you can do with it
Classroom inversion checks
Let students submit a sphere volume, surface area, or diameter and verify the single recovered radius against the cube-root, square-root, or half-length formula from the textbook.
Tanks, coatings, and inspection notes
Turn a vessel volume, a plated surface area, or a caliper diameter into the radius a drawing or stock-size calculation actually needs.
FAQ
What happens if the measurement is zero or negative?
A sphere radius is recovered only from a positive finite volume, surface area, or diameter; those failures are not billed.
Which units does the radius of a sphere calculator use?
Whatever unit you send, consistently. Cubic length for volume, square length for surface area, length for diameter. The radius comes back in that same length unit; there is no conversion.
How do I find the radius of a sphere from its volume?
Multiply the volume by three, divide by four π and take the cube root. A volume of thirty-six π, about 113.097335529233, gives a radius of 3 and therefore a diameter of 6.
How do I find the radius of a sphere from its surface area?
Divide the surface area by four π and take the square root. The radius comes out in the length unit that matches the area, so an area in square metres gives a radius in metres.
Why does the radius from volume need a cube root?
Because the volume grows with the cube of the radius: it is four thirds of π times r cubed. Undoing that takes a cube root, which is why doubling the volume does not double the radius.
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/math/radius-of-sphere \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"from":"volume","value":113.09733552923255}'const res = await fetch("https://api.kit.forhosting.com/math/radius-of-sphere", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"from": "volume",
"value": 113.09733552923255
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/radius-of-sphere",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"from": "volume",
"value": 113.09733552923255
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/radius-of-sphere", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"from":"volume","value":113.09733552923255}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"from":"volume","value":113.09733552923255}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/radius-of-sphere", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"from": "volume",
"value": 113.09733552923255
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.radius_of_sphere",
"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
The given measurement must be a positive finite number; zero, negatives, NaN, and infinities fail with invalid_input and are not billed.
The three inversions: volume, surface area, and diameter
Choose the field from as volume, surface_area, or diameter, and send the matching positive value.
Input contract, response shape, and what fails
On success the body returns radius, the recovered positive length rounded to twelve decimal places for stable goldens; from and value echoed so a client can display what it sent; formula, a short ASCII restatement of the inversion used; and diameter, which is always two times the recovered radius. Example: from volume and value equal to thirty-six pi (the volume of a sphere of radius three) yields radius three and diameter six. POST a JSON body with two required fields. from is an enum of volume, surface_area, or diameter — a closed set so the form can be a select and the API can reject typos instead of guessing. value is the given measurement and must be a positive finite number; numeric strings are accepted, empty strings and non-numeric text are not. Aliases given and from_measurement are accepted for from; measurement and given_value are accepted for value. Missing either required field, a from outside the enum, a non-finite value, or a value that is zero or negative all fail with invalid_input naming the field, and those failures do not consume balance. Pair the number with your own unit label; this endpoint does not emit plots, step-by-step prose beyond the formula field, or a conversion into another length system.
Integration use cases
Exam and LMS auto-grading
Mark radius-of-sphere items with a deterministic oracle that rejects non-positive measurements as invalid_input and never invents a second root.
Technical questions
How do I find the radius of a sphere from its volume?
Send from=volume and a positive finite value V. The API returns r = (3V/(4π))^(1/3), plus the matching diameter 2r.
Can I start from surface area or diameter instead?
Yes. from=surface_area inverts S = 4πr² as r = sqrt(S/(4π)). from=diameter returns r = d/2. Provide exactly one measurement kind in from.
What does finding the radius of a sphere cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.