Trirectangular tetrahedron calculator
A corner tetrahedron with perpendicular legs of 3, 4 and 5 has a volume of 10 and a total surface of 37.3654. Enter the three legs and get every face, edge and the space diagonal, with the slanted face from De Gua's theorem. Free in your browser.
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Worked example
- Perpendicular edge a
- 3
- Perpendicular edge b
- 4
- Perpendicular edge c
- 5
- Volume
- 10
- Areas of right-triangle faces
- 6, 7.5, 10
- Edges of the oblique face
- 5, 5.831, 6.4031
- Space diagonal
- 7.0711
Computes the geometry of a trirectangular tetrahedron with mutually perpendicular legs a, b, c: the volume abc/6, the three right-triangle face areas, the three hypotenuse edge lengths, the space diagonal, and the total surface area including the oblique face given by De Gua's theorem.
A corner cut from a box
A trirectangular tetrahedron is the solid you get at the corner of a rectangular box: three edges, a, b and c, meet at one vertex at right angles to each other. Three of its faces are right triangles that share that corner, and the fourth is a slanted triangle facing away from it. Because every measure follows from the three legs, those are the only values you enter. The volume is a·b·c divided by 6, and each right face is half the product of its two legs. With legs of 3, 4 and 5 the volume is 10 and the right faces measure 6, 7.5 and 10.
The slanted face and De Gua's theorem
The slanted face follows De Gua's theorem, the three-dimensional cousin of Pythagoras: the square of its area equals the sum of the squares of the three right faces. For legs of 3, 4 and 5 that sum is 192.25, so the slanted face measures about 13.8654 and the whole surface 37.3654. Working the slanted face out straight from the legs, instead of rebuilding its three sides first, keeps rounding error from piling up. The calculator also lists the three hypotenuse edges and the space diagonal. Every result is rounded once, to 4 decimals by default or to any number from 0 to 8.
The shape and its core formulas
A trirectangular tetrahedron has three edges a, b, and c meeting at one vertex at mutually right angles, exactly like the corner of a rectangular box. Every field is derived directly from the three legs — there is no other geometry to supply.
De Gua's theorem and the total surface area
The face opposite the right-angle vertex is an oblique triangle, and its area follows De Gua's theorem, the three-dimensional analogue of the Pythagorean theorem. De Gua states that the square of the oblique face's area equals the sum of the squares of the three right-triangle faces, so that area is (1/2)·sqrt((a·b)² + (a·c)² + (b·c)²). Computing the oblique face straight from the legs, rather than reconstructing its three side lengths first, keeps the result numerically clean and avoids compounding rounding error.
What you can do with it
Solid-geometry teaching
Demonstrate De Gua's theorem with a worked corner tetrahedron, showing how the oblique face relates to the three right faces.
CAD, meshing and FEM
Compute the volume and surface area of an orthogonal-leg corner wedge or a right-angled tetrahedral mesh element.
Material and print estimation
Get the exact volume abc/6 and total surface area to estimate material or coating for a right-corner solid part.
FAQ
What is a trirectangular tetrahedron?
A tetrahedron whose three edges a, b, c meet at one vertex at mutually right angles — the corner of a rectangular box.
How is the slanted face area found?
By De Gua's theorem: its area is (1/2)·sqrt((ab)²+(ac)²+(bc)²), and its square equals the sum of the squares of the three right-face areas.
How do I control precision?
The decimals field (default 4, range 0 to 8) rounds each output once, half-up, from the raw value; the volume abc/6 is rounded the same way.
How do I find the volume of a trirectangular tetrahedron?
Multiply the three perpendicular legs and divide by 6. Legs of 3, 4 and 5 give 60 divided by 6, a volume of 10, in the cube of whatever unit your legs are measured in.
What is the space diagonal of a trirectangular tetrahedron?
It is the length √(a² + b² + c²), the same as the diagonal of a box with those three edges. For legs of 3, 4 and 5 the calculator gives 7.0711 with the default 4 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
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/trirectangular-tetrahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":3,"b":4,"c":5}'const res = await fetch("https://api.kit.forhosting.com/math/trirectangular-tetrahedron", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 3,
"b": 4,
"c": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/trirectangular-tetrahedron",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 3,
"b": 4,
"c": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/trirectangular-tetrahedron", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":3,"b":4,"c":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":3,"b":4,"c":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/trirectangular-tetrahedron", 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": 3,
"b": 4,
"c": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.trirectangular_tetrahedron",
"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 shape and its core formulas
From those three legs this capability returns the volume, a·b·c/6; the three right-triangle faces that meet at the apex, with areas a·b/2, a·c/2, and b·c/2 (field face_areas_right); the three hypotenuse edges of those faces, sqrt(a²+b²), sqrt(a²+c²), and sqrt(b²+c²) (field hypotenuse_edges); and the space diagonal from the apex to the opposite face's far corner, sqrt(a²+b²+c²). With a=3, b=4, c=5 the volume is 10, the right-face areas are 6, 7.5, and 10, the hypotenuse edges are 5, 5.831, and 6.4031, and the space diagonal is 7.0711.
De Gua's theorem and the total surface area
The total_area field then sums all four faces: a·b/2 + a·c/2 + b·c/2 plus that oblique-face area. For a=3, b=4, c=5 the right faces are 6, 7.5, and 10, whose squares 36 + 56.25 + 100 = 192.25 give an oblique face of sqrt(192.25) ≈ 13.8654, so the total surface area is 37.3654.
Inputs, rounding and where it runs
Three positive finite numbers a, b, and c are required; any non-positive, non-finite, or non-numeric leg throws a typed invalid_input error that names the offending field. The optional decimals field defaults to 4 and must be an integer from 0 to 8, otherwise it is rejected with the same invalid_input error. Each rounded output — volume, every face area, every hypotenuse edge, the space diagonal, and total_area — is computed once with a half-up rule applied to the raw IEEE-754 value, never chained from an already-rounded number, so total_area stays consistent with its independently rounded parts. The pure-JavaScript core is deterministic, with no clock, no randomness, and no locale dependence, so the same three legs always return identical JSON.
Technical questions
What does total_area include?
All four faces: the three right triangles ab/2, ac/2, bc/2 plus the oblique De Gua face.
Is it free?
The on-device browser widget is free (tier A). The API costs $0.002 per successful task, and failed calls are not billed.