Tetrahedron volume calculator
Enter one edge and get every main measure of a regular tetrahedron at once: an edge of 2 gives a volume of 0.942809041582 and a surface of 6.928203230276, plus the height and the radii of the inner and outer spheres. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Edge
- 2
- Volume
- 0.9428090416
- Surface area
- 6.9282032303
- Face area
- 1.7320508076
- Height
- 1.6329931619
A regular tetrahedron is the simplest Platonic solid: four equilateral triangular faces, six equal edges, and four vertices. Given a single positive edge length a, closed-form formulas answer the measures people look up most often in school geometry, packing, and CAD pre-checks. Volume is a cubed times the square root of two, divided by twelve—the enclosed space of the solid. Total surface area is a squared times the square root of three, because four congruent equilateral faces each contribute one-quarter of that product. Height from a vertex to the opposite face is a times the square root of six over three. The insphere radius (inradius) is a times the square root of six over twelve—the radius of the largest sphere that sits inside and touches all four faces.
What volume, surface, height and insphere mean for a regular tetrahedron
A regular tetrahedron has four congruent equilateral triangular faces; every edge has the same length a, and every face angle is sixty degrees. The volume formula V equals a cubed times the square root of two over twelve is the classical closed form obtained by integrating triangular cross-sections or by placing the solid in coordinates with edge length a and computing one-sixth of the absolute scalar triple product of three edge vectors from a common vertex.
Surface area is four times the area of one equilateral triangle: each face has area square root of three over four times a squared, so the total surface is square root of three times a squared. Height H from a vertex straight down to the opposite face is a times square root of six over three; that altitude meets the face at its centroid, and the same geometry yields the inradius r equals a times square root of six over twelve—the distance from the tetrahedron centre to any face.
Units stay consistent with whatever length unit you pass for a: metres yield cubic metres of volume and square metres of area; inches yield cubic and square inches. A classic classroom check is a equals two: volume is two thirds times square root of two, surface is four times square root of three, height is two thirds times square root of six, and inradius is one sixth times square root of six.
The arithmetic is elementary multiplications and square roots after coercion, so there is no iteration, no Monte Carlo sampling, no dependence on wall-clock time, and no need for face coordinates or dihedral angles in the input. Related solids such as irregular tetrahedra with unequal edges, square pyramids, or triangular prisms need different formulas and are out of scope here.
Practical tips, limits, and related searches
When teaching, prefer paired examples: edge two against the classic hand-check values; edge one for the unit tetrahedron with volume square root of two over twelve and surface square root of three; and a zero or negative edge to demonstrate the hard rejection path. Prefer this tool when you already know the regular edge length and need volume plus surface, height, and inradius in one shot.
Round-trip checks are easy: after solving, confirm that surface area equals four times face area, that height equals three times the insphere radius times four over something—specifically that insphere radius equals height over four for a regular tetrahedron—and that circumradius equals three times the insphere radius. If you need only planar triangle area from three sides, use the Heron formula capability; if you need cylinder volume from radius and height, use the cylinder volume calculator instead of forcing a tetrahedron model.
What you can do with it
Packing, takeoff, and material estimates
Turn a regular tetrahedron edge into volume for fill or mass estimates, or total surface for coating, wrap, and material orders on tetrahedral stock.
CAD and geometry pipeline pre-checks
Validate regular tetrahedron dimensions in scripts before meshing or quoting: volume for density estimates, height and inradius for clearance and inscribed-sphere checks.
FAQ
What is the formula for the volume of a regular tetrahedron?
With edge length a, volume is (a³ · √2) / 12. The edge must be positive and use a consistent length unit; volume is then in the matching cubic unit.
What is the surface area of a regular tetrahedron?
Total surface area is a² · √3 (four equilateral faces). One face has area (√3 / 4) · a².
How are height and insphere radius related?
Height from a vertex to the opposite face is (a · √6) / 3. Insphere radius is (a · √6) / 12, which is one quarter of that height.
Why is a zero or negative edge rejected?
A solid regular tetrahedron needs a strictly positive edge.
Is the tetrahedron volume calculator free?
The browser tool is free and runs locally.
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/tetrahedron \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"edge":2}'const res = await fetch("https://api.kit.forhosting.com/geom/tetrahedron", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"edge": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/tetrahedron",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"edge": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/tetrahedron", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"edge":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"edge":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/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
{
"edge": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.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.
Limits
max_edge | 1000000000000 |
min_edge | 1e-15 |
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 capability packages those identities as deterministic JSON with no computer-algebra system, no random seeds, and the same pure solve function in the free browser tool and the prepaid API path. A non-positive edge is rejected with invalid_input so scripts and grading pipelines see an honest error instead of a silent absolute value or a zero-volume solid.
What volume, surface, height and insphere mean for a regular tetrahedron
Zero or negative edge length is not accepted under this contract because it does not describe a solid regular tetrahedron of positive extent; the API returns invalid_input rather than inventing a magnitude.
How to call the API and what comes back
The value must be a finite number strictly greater than zero and inside the published limits up to one trillion. Use it from geometry homework pipelines, packing and takeoff scripts, interview-prep sandboxes, CAD pre-checks, or any workflow that needs reproducible regular-tetrahedron numbers without installing a computer-algebra package on every machine. Send a JSON body with a single required field edge—the positive edge length of the regular tetrahedron. Aliases a, side, edge_length, and length are accepted and map to the same meaning. On success the response echoes edge; returns volume, surface_area, height, and insphere_radius as the primary results; and includes face_area for one equilateral face, circumradius of the circumscribed sphere, method set to regular_tetrahedron, and a short formula block documenting the closed forms. Failed validation—missing the required field, a non-finite number, a non-positive edge, or a value outside published limits—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price. Pair the call with unit conversion only when your source data mixes millimetres and metres; this capability does not convert linear units for you, so keep the edge in one consistent unit before you send the request and interpret volume and area in the matching cubic and square units.
Practical tips, limits, and related searches
Keep the edge length inside the published positive range up to one trillion so intermediate cubes and square roots stay finite in IEEE double precision on the Worker and in the browser. Because the formulas only need a, students can recompute by hand for nice radicals and match the API line by line after comparing reported floats to exact expressions involving square roots of two, three, and six. This landing targets people searching for a tetrahedron volume calculator or regular tetrahedron surface area, height, and insphere radius from edge length; it is not a general polyhedron mesh tool or an irregular tetrahedron solver. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible measures while learning the contract.
Integration use cases
Homework checks for regular tetrahedron measures
Give students an edge length; compare their hand-computed volume, surface area, height, and inradius against a deterministic API or free browser result.