Oblique prism volume
An oblique prism with a base area of 12 and a perpendicular height of 5 holds 60 cubic units, just like an upright prism with the same base and height. Enter the base area and the height between the bases to get the volume. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Base area
- 12
- Height
- 5
- Volume
- 60
An oblique prism has two congruent parallel bases connected by parallelogram faces that are not necessarily perpendicular to the bases. The solid may lean, yet its volume is still the product of the base area and the perpendicular distance between those bases. Cavalieri's principle makes this precise: if every plane parallel to a fixed reference cuts two solids in regions of equal area, the solids have equal volume. An oblique prism and a right prism that share the same base area and the same orthogonal height therefore share the same volume formula V equals B times h. Students meet this result when a textbook shows a slanted figure and asks for volume without expanding every lateral edge. Architects, packaging designers, and CAD helpers need the same number when a drawing lists base area and story height rather than a full mesh.
What an oblique prism is and why Cavalieri fixes the volume
In Euclidean solid geometry a prism is a polyhedron with two parallel congruent polygonal bases and lateral faces that are parallelograms. When the lateral edges are perpendicular to the bases, the prism is right; when they are not, it is oblique. The visual lean can trick the eye into thinking volume changed, but the measure that matters for volume is the perpendicular distance between the two base planes, not the length of a slanted lateral edge.
Cavalieri's principle, named after Bonaventura Cavalieri, states that solids with equal cross-sectional area at every corresponding height have equal volume. Cutting an oblique prism with planes parallel to the bases produces slices congruent to the base, just as in the right case, so the stack of infinitesimal slices has the same total volume. Algebraically this collapses to the elementary product V equals B times h, where B is the area of either base and h is that orthogonal height.
Search intent clusters around phrases like oblique prism volume, Cavalieri prism calculator, base area times height, and perpendicular height of a prism. People want a numeric volume they can check against a homework key or a bill of materials, not only a geometric argument that the solid is determined.
What you can do with it
Homework and exam oblique-prism drills
Students and graders confirm V=B·h for slanted prisms from base area and perpendicular height, including rejection when either value is not positive.
FAQ
Which formula does the calculator use?
Cavalieri's principle: volume equals base area times perpendicular height (V = B · h). An oblique prism has the same volume as a right prism with the same base area and the same orthogonal height.
Is the oblique prism volume calculator free?
The browser tool runs free on your device.
Can I use a slanted lateral edge as the height?
No. Height must be the perpendicular distance between the two parallel bases. If you only know a slanted edge, project it to the orthogonal height first; Cavalieri requires that perpendicular measure.
Why doesn't the lean change the volume of a prism?
By Cavalieri's principle, solids with equal cross-sections at every height have equal volume. Every slice of an oblique prism parallel to its bases is a copy of the base, exactly as in an upright prism, so base area times height still holds.
Which units should I use?
Any consistent pair. A base area in square centimetres and a height in centimetres give a volume in cubic centimetres, with no conversion along the way, so keep both measures in the same unit system.
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/oblique-prism \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"base_area":12,"height":5}'const res = await fetch("https://api.kit.forhosting.com/geom/oblique-prism", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"base_area": 12,
"height": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/oblique-prism",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"base_area": 12,
"height": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/oblique-prism", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"base_area":12,"height":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"base_area":12,"height":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/oblique-prism", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"base_area": 12,
"height": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.oblique_prism",
"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 accepts strictly positive base_area and height, rejects zero or negative measures, and returns volume with the Cavalieri method tag as structured JSON.
What an oblique prism is and why Cavalieri fixes the volume
Delivering volume, base_area, height, method, and formula as JSON makes the result easy to paste into notes, diff in automated tests, or feed into a larger pipeline without reimplementing the product in every language you teach or deploy across lesson platforms and packaging tools.
How this calculator validates input and applies V equals B times h
Both must be positive finite numbers. Edge cases are explicit: missing fields, non-numeric strings, and non-positive measures all fail before any volume is returned. You may use any consistent unit system; if base area is in square centimeters and height in centimeters, volume is in cubic centimeters without conversion inside the engine. Do not pass a slanted lateral edge as height unless you have already projected it to the orthogonal distance between the bases, because Cavalieri requires the perpendicular measure, not the edge length of a leaning side face. Send a JSON body with two required fields: base_area and height. The engine parses each field as a number or a numeric string, rejects empty or non-finite values, and then enforces strict positivity: zero and negative measures fail with an invalid_input message that names the bad field. When both measures pass, volume is the product of base_area and height. Results are rounded to a stable decimal precision so the same input always yields the same JSON across JavaScript engines. The response echoes the validated base_area and height, tags method as cavalieri, and records the formula string V equals B times h for transparency in lesson materials.
Practical use, limits, pricing, and delivery channels
Use the browser tool when you want an immediate oblique-prism volume while checking solid-geometry homework, verifying a leaning column volume from base area and story height, or confirming a packaging sketch 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. Remember the contract limits: this tool does not compute lateral or total surface area, does not convert units, and does not accept a slanted edge in place of perpendicular height. Prefer explicit domain errors over opaque black-box widgets that cannot show why a non-positive measure is rejected when students paste zero from a partially filled form. Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs must be strictly positive; zero-area bases and non-positive heights are rejected with a clear invalid_input message. Results are fully deterministic: the same base_area and height always yield the same volume. Wire the endpoint into solid-geometry drills, contest answer checkers, CAD preprocessing steps that already know base area and orthogonal height, or floating worksheets that list only those two measures.
Integration use cases
Deterministic fixtures for solid-geometry tests
Pipelines that seed unit tests or interactive prism demos can call a pure Cavalieri oracle instead of reimplementing the product in every language.
Architecture and packaging from base area and story height
Notes that list only base area and orthogonal height recover volume as structured JSON for reports, size charts, or lightweight CAD helpers without full meshes.
Technical questions
How do I compute the volume of an oblique prism with this API?
Send a JSON body with positive base_area and height to geom.oblique_prism. The response returns volume (base_area×height), the echoed inputs, method cavalieri, and the formula V = B · h.
What happens if base area or height is not positive?
The request fails with invalid_input. Both base_area and height must be strictly positive finite numbers; zero and negative values are rejected and those calls are not charged.