Square pyramid volume
Solve a square pyramid from just its base edge and vertical height. An edge of 6 and a height of 4 give a volume of 48, a slant height of 5 and a total surface of 96, base included. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Base edge
- 6
- Height
- 4
- Base area
- 36
- Volume
- 48
- Slant height
- 5
- Lateral surface area
- 60
This calculator turns a positive square base edge and perpendicular height into the classic square-pyramid metrics you need for homework, design, and code: the exact volume one-third base-edge squared times height, the face slant height as the hypotenuse of height and half the base edge, the lateral surface area as two times base edge times that slant, and the total surface that adds the square base.
What a square pyramid calculator returns and why base edge and height matter
A square pyramid is a polyhedron with a square base of side length a and four congruent isosceles triangular faces that meet at a single apex directly above the center of the base when the solid is right. The perpendicular distance from that apex down to the base plane is the height h, and the classic volume formula is always one-third of the product of the base area a squared and that height.
This tool exists to turn those two positive lengths into a single deterministic response without forcing you to look up which third of the prism volume applies, how face slant height is defined from the midpoint of a base edge up to the apex, or whether total surface includes the base.
If you need a general pyramid from base area alone, a cone from radius and height, or a rectangular base, use a tool dedicated to those problems; the search intent here is square pyramid volume from base edge and height, with face slant and surface areas, and a clear answer or an honest error when the dimensions are not positive.
How the closed forms are evaluated: volume, face slant, lateral and total surface
Volume is the exact product of one-third, base edge squared, and height, independent of any approximation or sampling. Face slant height is the altitude of one lateral triangular face measured from the midpoint of a base edge up to the apex: the square root of height squared plus half the base edge squared, which is exactly five when base edge is six and height is four, a convenient check that mirrors the classic three-four-five pattern after the half-edge of three.
Lateral surface area multiplies two, the base edge, and that slant height, equal to four times the area of one isosceles face triangle with base a and face height l. Base area is the ordinary square a squared. Total surface area is the sum of base and lateral surfaces, equivalently a squared plus two a l.
Optional precision only rounds the published fields; the internal solve always uses full double precision before that final rounding step.
What you can do with it
Homework and exam preparation
Compute square pyramid volume, face slant height, lateral surface, and total surface from textbook base edge and height, and verify that non-positive dimensions are rejected.
Roofs, monuments, and packaging design
Size pyramidal roof volume, monument mass from density, or packaging pyramid material from known square base edge and height in CAD and manufacturing tools.
FAQ
What does the square pyramid volume calculator cost?
It is free in your browser on this page.
How is volume computed?
Volume is always V = (1/3) · a² · h, where a is the square base edge and h is the perpendicular height.
How are slant height and surface areas defined?
Face slant height is √(h² + (a/2)²). Lateral surface is 2 · a · l (four triangular faces). Total surface is a² + 2 · a · l.
What are the results for a base edge of 6 and a height of 4?
The base area is 36, the volume 48, the slant height 5, the lateral surface 60 and the total surface 96. The slant comes out whole because half the edge, 3, and the height, 4, form a 3-4-5 triangle.
Should I enter the vertical height or the slant height?
Enter the vertical height, the perpendicular distance from the apex down to the base. The calculator works out the slant height of each face from it and from half the base edge.
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/square-pyramid-solver \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"base_edge":6,"height":4}'const res = await fetch("https://api.kit.forhosting.com/geom/square-pyramid-solver", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"base_edge": 6,
"height": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/square-pyramid-solver",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"base_edge": 6,
"height": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/square-pyramid-solver", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"base_edge":6,"height":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"base_edge":6,"height":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/square-pyramid-solver", 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": 6,
"height": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.square_pyramid_solver",
"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
You enter base_edge and height as finite positive lengths; the engine returns base_edge, height, base_area, volume, slant_height, lateral_surface_area, total_surface_area, and formula strings so a student or a log line can verify the identity without reading the source. When base edge or height is missing, zero, negative, or non-finite, the tool returns invalid_input instead of a silent NaN. Through the API a successful request costs $0.002; validation failures are not charged.
What a square pyramid calculator returns and why base edge and height matter
The response always echoes base_edge and height, reports base_area as a squared, volume as one-third a squared h, slant_height as the square root of h squared plus half-edge squared, lateral_surface_area as two times a times that slant, total_surface_area as base plus lateral, and formula strings for volume, slant height, lateral surface, and total surface so a student or a log line can verify the identity without reading the source. Missing base edge or height, non-finite values, or a non-positive dimension all raise invalid_input with a message that names the failing field. That discipline keeps automated tests stable, makes caching safe, and prevents a script from treating a degenerate solid as a real square pyramid.
How the closed forms are evaluated: volume, face slant, lateral and total surface
The engine resolves base_edge from the field base_edge with aliases side, a, base_side, and edge, and height from height with aliases h and H, requiring each to be finite and strictly greater than zero. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the one-third volume rule or the face-slant hypotenuse yourself for every square pyramid in a homework set or a batch job.
Where square pyramid metrics fit in study, design, and code
Students meet the square pyramid when a textbook asks for volume from base edge and height, when an exam demands lateral area given the face slant height, or when solid-geometry homework wants total surface including the base before painting or material estimates. Teachers can pre-generate answer keys that list volume and surfaces for integer edges and heights, and that refuse zero or negative dimensions instead of printing NaN. Architecture and civil design often need the capacity of a pyramidal roof volume, the material area of four triangular faces, or the mass of a monument from known base edge and height; knowing that a requested height is not positive is as valuable as the number itself. Surveying and CAD workflows place pyramid volumes as construction geometry, and game or simulation teams use volume and surface for collision approximations, particle emitters, and procedural mesh sizing that should stay deterministic across platforms. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with other free geometry endpoints when you need cone volume, ellipse metrics, or triangle solvers for a lab of deterministic tools, and with documentation that shows both dimensions and the finished metrics so the numbers and the picture stay aligned for learners and reviewers. Whether you arrived searching for square pyramid volume or for slant height and surface area from base edge and height, the contract is the same: positive base edge and height in, finite metrics or a clear invalid_input out.
Integration use cases
Deterministic geometry fixtures
Generate reproducible square-pyramid metrics for unit tests, contest validators, and CI assertions without hand-rolled one-third volume or slant-height logic.
Technical questions
What if base edge or height is zero or negative?
The handler returns invalid_input with a message that the field must be a positive finite number.
Can I control rounding?
Yes. Set precision (aliases decimals, dp) to an integer from 0 to 15; the default is 10 decimal places on all numeric fields.