Triangular pyramid calculator
Get the volume and surface area of a regular triangular pyramid from just two measurements: a base edge of 2 and a height of 3 give a volume of 1.732050807569 and a total surface of 10.897202197481. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Base edge
- 2
- Height
- 3
- Base area
- 1.7320508076
- Volume
- 1.7320508076
- Lateral surface area
- 9.1651513899
- Surface area
- 10.8972021975
A regular triangular pyramid has an equilateral triangle as its base and three congruent isosceles triangular faces that meet at an apex directly above the center of that base. Given the common base edge length and the perpendicular height from apex to base plane, this calculator returns the enclosed volume, the triangular base area, the mid-edge face slant height, the lateral surface area of the three side faces, and the total surface area (base plus lateral) with the classic closed-form formulas used in solid-geometry textbooks.
Where triangular pyramid calculations show up in real work
Secondary-school and college solid-geometry courses constantly ask students for the volume and surface area of a regular triangular pyramid given edge and height; a trusted calculator with a strict positive-input gate speeds homework grading and removes spreadsheet mistakes where someone uses a square-base formula or forgets the equilateral base area.
Architecture, roofing and fabrication workflows size triangular tent peaks, pavilion corners, monument caps, and decorative planters from plan edge length and cut height; volume feeds fill and concrete estimates while total surface area feeds cladding and paint takeoffs before lateral faces are detailed. Packaging, display stands and product design often use regular triangular pyramids for point-of-sale geometry; deterministic metrics support quotes without a full mesh library.
Game, graphics and CAD libraries store regular pyramid primitives for collision proxies and lighting volumes; echoing method plus the two inputs makes debug property panels self-describing. If you later need a square or hexagonal pyramid, a general pyramid from base area only, a triangular prism, or a regular tetrahedron constrained so all four faces are equilateral, treat those as separate search intents rather than overloading this triangular-pyramid volume landing.
What you can do with it
Homework and exam checks
Compute regular triangular pyramid volume and surface area from edge and height, and reject negative or non-positive inputs with a clear error.
Roofs, tents and fabrication
From plan edge and apex height, recover fill volume and cladding area for triangular peaks, planters and monument caps.
CAD and game pyramid metrics
Turn base edge and height into stable volume and surface figures for layout, material estimates and property panels.
FAQ
What formulas does this calculator use?
Base area = (√3/4) a²; volume = (√3/12) a² h; slant height m = √(h² + a²/12); lateral surface = (3/2) a m; total surface = base + lateral.
Is the apex assumed to be centered above the base?
Yes. This is a regular triangular pyramid: the base is an equilateral triangle and the apex lies on the axis through the center of that base, perpendicular to the base plane.
What does it cost?
The browser tool is free and runs locally.
What is the slant height of a triangular pyramid?
It is the height of a side face, measured from the middle of a base edge up to the apex. With a base edge of 2 and a height of 3, the slant height is 3.055050463304.
Is this the same as a regular tetrahedron?
Not necessarily. Here only the base is equilateral and you choose the height freely; a regular tetrahedron requires all four faces to be equilateral, which is a separate calculation.
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/triangular-pyramid \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"base_edge":2,"height":3}'const res = await fetch("https://api.kit.forhosting.com/geom/triangular-pyramid", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"base_edge": 2,
"height": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/triangular-pyramid",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"base_edge": 2,
"height": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/triangular-pyramid", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"base_edge":2,"height":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"base_edge":2,"height":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/triangular-pyramid", 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_edge": 2,
"height": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.triangular_pyramid",
"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
Any non-positive edge or height—including every negative dimension—is rejected with a clear invalid_input error instead of a silent nonsense volume. No network calls, no random sampling, and no floating-point surprises beyond ordinary IEEE-754 arithmetic on the numbers you provide.
How volume and surface area are computed
Pyramid volume is one-third of base area times height, which simplifies for this base to volume equals (√3/12) a² h. For surface area, first form the base inradius (apothem of the equilateral triangle) r equals a √3 / 6; the face slant height measured from the midpoint of a base edge up the altitude of a lateral face is m equals the square root of h squared plus r squared, which is √(h² + a²/12). Lateral surface area is half the base perimeter times that slant: (1/2)×3a×m equals (3/2) a m. Because the path uses only multiplication, division, and square roots of three and of a sum of squares, it is deterministic, O(1), and free of network or random sampling. Label the side length of the equilateral triangular base as base_edge a and the perpendicular distance from the apex to the base plane as height h. The area of an equilateral triangle is one-fourth times the square root of three times a squared: base_area equals (√3/4) a². Total surface area is base_area plus lateral_surface_area. Classic checks include a equals two and h equals three: base_area equals √3, volume equals √3, and the slant and lateral terms follow from √(h² + a²/12) without iterative solvers. Huge magnitudes can still overflow IEEE-754 double range; when any result is non-finite the request fails with a clear overflow message instead of returning Infinity in JSON. That closed form matches solid-geometry textbooks and keeps automated graders aligned with hand computation on the same inputs for every positive pair you submit.
What you send and what you get back
There is no optional circumradius, lateral edge, apex offset, or irregular triangle sides—this landing is the regular centered triangular pyramid from edge and height only. Example checks: edge two and height three recover √3 multiples for base and volume; edge one and height one yield volume √3/12 and a surface area that includes both the triangular base and three isosceles lateral faces. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing triangular-pyramid formulas in every language. Provide two required finite numbers: base_edge is the side length of the equilateral triangular base and must be strictly positive; height is the perpendicular apex-to-base distance and must also be strictly positive. Zero, negative, missing, empty, non-numeric, NaN, and Infinity values are rejected with invalid_input and are not charged—negative dimensions always fail the gate as the product contract requires. On success the JSON response returns base_edge, height, base_area, volume, lateral_surface_area, surface_area, slant_height, and method set to regular_triangular_pyramid_volume_surface so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where triangular pyramid calculations show up in real work
Because this capability is a pure parse engine, failed validation does not consume balance and successful runs stay at $0.002 per item.
Technical questions
What happens if base_edge or height is zero or negative?
The request fails with invalid_input. Both values must be strictly greater than zero; negative dimensions are never accepted. The error message names the field (base_edge or height).
Does surface_area include the base?
Yes. surface_area is the total surface area (triangular base plus three lateral faces). lateral_surface_area is the three triangles alone if you need only the sides.