Octagonal pyramid calculator
With base side 2 and height 3, a regular octagonal pyramid holds exactly its base area in volume, since a pyramid is a third of base times height. Enter side and height for the volume, slant height and surface of roofs, planters or homework. 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
- 19.313708499
- Volume
- 19.313708499
- Lateral surface area
- 30.8061574362
- Surface area
- 50.1198659352
A regular octagonal pyramid has a regular octagon as its base and eight congruent isosceles triangular faces that meet at an apex directly above the center of that base. Given the common base side length and the perpendicular height from apex to base plane, this calculator returns the enclosed volume, the octagonal base area, the mid-edge face slant height, the lateral surface area of the eight triangles, and the total surface area (base plus lateral) with the classic closed-form formulas used in solid-geometry textbooks.
Where octagonal 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 octagonal pyramid given side and height; a trusted calculator with a strict positive-input gate speeds homework grading and removes spreadsheet mistakes where someone uses a square or hexagonal base formula or forgets the geometric base area of the octagon.
Architecture, roofing and fabrication workflows size octagonal tent peaks, gazebo roofs, monument caps, and decorative planters from plan side 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 octagonal 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 pyramid frustum, a square, hexagonal or triangular pyramid, or only base area without the solid, treat those as separate search intents rather than overloading this octagonal-pyramid landing.
What you can do with it
Homework and exam checks
Compute regular octagonal pyramid volume and surface area from side and height, and reject negative or non-positive inputs with a clear error.
Roofs, gazebos and fabrication
From plan side and apex height, recover fill volume and cladding area for octagonal peaks, planters and monument caps.
CAD and game pyramid metrics
Turn base side and height into stable volume and surface figures for layout, material estimates and property panels.
FAQ
Is the apex assumed to be centered above the base?
Yes. This is a regular octagonal pyramid: the base is a regular octagon 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.
How is the area of the octagonal base worked out?
A regular octagon with side a has an area of 2(1 + √2) times a squared. With a side of 2 that gives 19.313708498985 square units, the starting point for both the volume and the total surface.
What is the slant height of an octagonal pyramid?
It is the distance from the middle of a base edge up a triangular face to the apex. It combines the height with the octagon's apothem; for side 2 and height 3 it is 3.850769679525. The lateral area is four times the side times this slant height.
Which result do I use for concrete, cladding or paint?
Use the volume for fill or concrete, the lateral area when only the eight sloping faces get covered, and the total surface area when the octagonal base is clad or painted as well.
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/octagonal-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/octagonal-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/octagonal-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/octagonal-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/octagonal-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.octagonal_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 negative or non-positive side or height 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 (2/3)(1+√2) a² h. For surface area, first form the base apothem (inradius of the octagon) r equals a(1+√2)/2; 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. Lateral surface area is half the base perimeter times that slant: (1/2)×8a×m equals 4 a m. Because the path uses only multiplication, division, and square roots of two and of a sum of squares, it is deterministic, O(1), and free of iterative solvers. Label the side length of the regular octagonal base as base_edge a and the perpendicular distance from the apex to the base plane as height h. The area of a regular octagon is twice one plus the square root of two times a squared: base_area equals 2(1+√2) a². Total surface area is base_area plus lateral_surface_area. Classic checks include a equals two and h equals three: base_area equals 8(1+√2), volume equals 8(1+√2), and the slant, lateral, and total surface follow from the apothem and mid-edge face altitude. 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.
What you send and what you get back
There is no optional circumradius, lateral edge, or apex offset—this landing is the regular centered octagonal pyramid from side and height only. Example checks: side two and height three recover the (1+√2) multiples above; side one and height one yield volume (2/3)(1+√2) and a surface area that includes both the octagonal base and eight isosceles triangles. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing octagonal-pyramid formulas in every language. Provide two required finite numbers: base_edge is the side length of the regular octagonal 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. On success the JSON response returns base_edge, height, base_area, volume, lateral_surface_area, surface_area, slant_height, and method set to regular_octagonal_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 octagonal 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 formulas does this calculator use?
Base area = 2(1+√2) a²; volume = (2/3)(1+√2) a² h; apothem r = a(1+√2)/2; slant height m = √(h² + r²); lateral surface = 4 a m; total surface = base + lateral. The response includes method: "regular_octagonal_pyramid_volume_surface".
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 not allowed. The error message names the field (base_edge or height).
Does surface_area include the base?
Yes. surface_area is the total surface area (octagonal base plus eight lateral triangular faces). lateral_surface_area is the eight triangles alone if you need only the sides.