Spherical wedge volume calculator
Find how much of a solid ball lies between two half-planes that share a diameter: on a sphere of radius 5, a 90° opening holds 130.8996938996 cubic units, one quarter of the ball. Enter angles from 0 to 360 degrees. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 5
- Angle
- 90
- Angle radians
- 1.5707963268
- Volume
- 130.8996938996
A spherical wedge is the portion of a solid sphere cut by two half-planes that share a diameter of the sphere. Those planes form a dihedral angle, and the wedge volume is simply that angle’s share of the full ball: one full turn (360 degrees or 2π radians) recovers the classic four-thirds pi R cubed sphere volume, while a right angle recovers one quarter of the ball. Only two measures are required—a positive sphere radius R and a dihedral angle in degrees between zero and three hundred sixty inclusive. Students meet the identity in solid-geometry units after sphere volume; engineers use it when a tank, globe, or packaging sketch is described by radius and opening angle rather than a full CAD solid.
What a spherical wedge is and which two measures determine it
In Euclidean solid geometry a spherical wedge is the solid region swept by radii from the sphere center that lie between two fixed half-planes meeting along a diameter. The sphere radius R is the distance from center to surface. The dihedral angle α is the angle between those half-planes, measured in degrees for this capacity and converted internally to radians as θ equals α times pi over one hundred eighty.
Because a full sphere is the special case α equals three hundred sixty degrees, the wedge volume is exactly the fraction α over three hundred sixty of the full ball volume four-thirds pi R cubed. Algebraically that is the same as V equals two-thirds R cubed times θ. When α is ninety degrees the volume is one quarter of the sphere; when α is one hundred eighty degrees it is half the sphere; when α is zero the solid degenerates to zero volume along the diameter alone.
Search intent clusters around phrases such as spherical wedge volume calculator, volume of a spherical wedge, dihedral angle sphere volume, sphere sector by angle, and spherical wedge radius and angle. People want numeric results they can check against an answer key or a tank sketch, not only a geometric argument that the solid is determined.
How this calculator validates input and applies the closed-form formulas
Radius must be a strictly positive finite number within a stable numeric range. When both checks pass, the engine converts the angle to radians, multiplies two-thirds times R cubed times that radian measure, and returns the rounded volume together with the recovered inputs and the angle in radians. Edge cases are explicit: missing fields, non-numeric strings, zero or negative radius, and angles outside zero to three hundred sixty all fail before any metric is returned.
Boundary values zero and three hundred sixty are accepted: zero yields volume zero, and three hundred sixty recovers the full sphere volume four-thirds pi R cubed within floating precision. You may use any consistent unit system for radius; volume is expressed in the corresponding cubic units without conversion inside the engine.
What you can do with it
Homework and exam spherical-wedge drills
Students and graders confirm wedge volume from radius and dihedral angle in degrees, including rejection when the angle lies outside the closed 0–360° interval.
FAQ
Which formula does the calculator use?
Volume is (2/3)·R³·θ with θ = angle·π/180 in radians. Equivalently, V = (angle/360)·(4/3)·π·R³, the angle’s share of the full sphere volume.
What happens if the angle is outside 0–360 degrees?
A spherical-wedge dihedral angle must lie between 0 and 360 degrees inclusive; values outside that range are rejected and those calls are not charged.
Is the spherical wedge volume calculator free?
The browser tool runs free on your device.
How is a spherical wedge different from a rectangular wedge or a spherical sector?
A rectangular wedge uses base length, base width, and two end heights. A spherical sector uses radius and cap height (cone from center plus cap). A spherical wedge uses radius and the dihedral angle between two half-planes through a diameter.
What do angles of 0 and 360 degrees give?
Both are accepted. At 0 degrees the wedge shrinks to the diameter alone and its volume is zero; at 360 degrees it becomes the whole ball, with the full volume (4/3)πR³.
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-wedge \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5,"angle":90}'const res = await fetch("https://api.kit.forhosting.com/geom/spherical-wedge", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 5,
"angle": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/spherical-wedge",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 5,
"angle": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/spherical-wedge", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":5,"angle":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":5,"angle":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/spherical-wedge", 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,
"angle": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.spherical_wedge",
"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
This capacity applies the closed form V equals two-thirds R cubed times theta (theta in radians), returns radius, angle, angle in radians, and volume as structured JSON, and rejects any angle outside the zero-to-three-hundred-sixty degree interval with an explicit invalid_input error.
What a spherical wedge is and which two measures determine it
Delivering radius, angle, angle_radians, volume, and a fixed method label as JSON makes the result easy to paste into notes, pin in automated tests, or feed into a larger pipeline without reimplementing pi multiplications and degree-to-radian conversion in every language you teach or deploy.
How this calculator validates input and applies the closed-form formulas
Send a JSON body with both required fields radius and angle. Angle must be a finite number that lies in the closed interval from zero through three hundred sixty degrees inclusive; if the angle is less than zero or greater than three hundred sixty, the request fails with invalid_input and a message that names the zero-to-three-hundred-sixty bound, because a dihedral angle outside a full turn is not a valid Euclidean spherical wedge under this contract. Results use a stable decimal rounding so the same input always yields the same JSON across JavaScript engines.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate spherical-wedge volume while checking homework, sizing a globe segment opened by a known dihedral angle, verifying a storage or packaging volume from a measured radius and opening angle, or confirming a workshop drawing without installing a computer-algebra system. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials. Do not confuse this tool with a rectangular wedge calculator, which uses base length, base width, and two end heights, or with a spherical sector calculator, which uses cap height rather than a dihedral angle through a diameter. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs require positive radius and an angle between zero and three hundred sixty degrees inclusive; angles outside that range are always rejected with invalid_input, and zero or non-finite radius is likewise refused. Results are fully deterministic: the same radius and angle always yield the same angle_radians and volume. Wire the endpoint into solid-geometry drills, contest answer checkers, tank capacity worksheets that list only R and dihedral angle, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why an angle outside zero to three hundred sixty is rejected.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive sphere demos can call a pure spherical-wedge oracle instead of reimplementing (2/3)R³θ and degree-to-radian conversion in every language.
Tanks, globes, and packaging solids from R and opening angle
Shop or logistics notes that list sphere radius and dihedral opening angle recover wedge volume as structured JSON for reports, capacity charts, or lightweight CAD helpers.
Technical questions
How do I compute spherical wedge volume with this API?
Send a JSON body with positive radius and angle (degrees, 0–360) to geom.spherical_wedge. The response returns radius, angle, angle_radians, volume, and method.