Cone volume calculator
Enter the base radius and height of a cone and get its volume, slant height and surface areas: radius 3 and height 4 give a slant height of exactly 5 and a volume of about 37.7. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 3
- Height
- 4
- Slant height
- 5
- Volume
- 37.6991118431
- Base area
- 28.2743338823
- Lateral surface area
- 47.1238898038
This calculator turns a positive base radius and perpendicular height into the classic right circular cone metrics you need for homework, design, and code: the exact volume one-third pi r squared h, the slant height as the hypotenuse of r and h, the lateral surface area as pi r times that slant, and the total surface that adds the circular base.
Everything a cone has, from two lengths
A right circular cone is what you get by spinning a right triangle around one of its legs: the base is a circle of radius r and the tip sits straight above its center at height h. Those two lengths fix everything else. The volume is one third of π r² h, a third of the cylinder with the same base and height. The slant height is the straight line from the tip to the rim, the square root of r² plus h², so a radius of 3 and a height of 4 give exactly 5. The lateral area is π times r times the slant height, the same as the flat sector you would get by unrolling the side. Add the base, π r², and you have the total surface.
Where it helps and what usually goes wrong
It sizes the capacity of a hopper, the wall of a funnel or the material for a packaging cone, and it checks homework in seconds. The classic slips are forgetting the one third in the volume, mixing up the slant height with the perpendicular height measured from the tip to the center of the base, and not knowing whether a surface figure includes the base. The calculator gives the lateral and the total surface separately, so you pick the one you need. A radius or height of zero or less is rejected, and results are rounded to the decimals you choose, ten by default.
How the closed forms are evaluated: volume, slant, lateral and total surface
Volume is the exact product of one-third, pi, radius squared, and height, independent of any approximation. Slant height is the Euclidean length of the generating line: the square root of radius squared plus height squared, which is exactly five when radius is three and height is four, the classic Pythagorean triple that textbooks use to check mental arithmetic. Lateral surface area multiplies pi, radius, and that slant height, equal to the area of the sector you would get if you unrolled the side into a flat net.
Base area is the ordinary circle pi r squared. Total surface area is the sum of base and lateral surfaces, equivalently pi r times the sum of radius and slant height. 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 cone volume, slant height, lateral surface, and total surface from textbook radius and height, and verify that non-positive dimensions are rejected.
Hoppers, funnels, and packaging design
Size conical hopper capacity, funnel wall area, or packaging cone material from known base radius and height in CAD and manufacturing tools.
FAQ
What does the cone volume calculator cost?
It is free in your browser on this page.
What is the formula for the volume of a cone?
One third of π times the radius squared times the height, which is a third of a cylinder with the same base and height. A radius of 3 and a height of 4 give about 37.7 cubic units.
How is the slant height of a cone found?
It is the straight line from the tip to the rim of the base, the square root of radius squared plus height squared. With a radius of 3 and a height of 4 it is exactly 5.
What is the difference between lateral and total surface area?
The lateral area covers only the sloping side, π times radius times slant height, about 47.1 for radius 3 and height 4. The total adds the round base and comes to about 75.4.
Does it work for an oblique cone?
No. It handles the right circular cone, with the tip directly above the center of the base. For a cone whose tip leans to one side, use a tool made for oblique cones.
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/cone \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":3,"height":4}'const res = await fetch("https://api.kit.forhosting.com/geom/cone", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 3,
"height": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/cone",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 3,
"height": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/cone", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":3,"height":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":3,"height":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/cone", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius": 3,
"height": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.cone",
"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 radius and height as finite positive lengths; the engine returns radius, height, slant_height, volume, base_area, 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 radius 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 cone calculator returns and why radius and height matter
A right circular cone is the solid formed by rotating a right triangle about one leg: the base is a circle of radius r, the apex sits directly above the center at perpendicular height h, and the straight line from apex to any point on the base rim is the slant height l. This tool exists to turn those two positive lengths into a single deterministic response without forcing you to look up which formula uses one-third of the cylinder volume, how slant height relates to the Pythagorean theorem, or whether total surface includes the base. If you need cylinder volume, sphere surface, or ellipse metrics, use a tool dedicated to those problems; the search intent here is the right circular cone from radius and height and a clear answer or an honest error when the dimensions are not positive. The response always echoes radius and height, reports slant_height as the square root of r squared plus h squared, volume as one-third pi r squared h, base_area as pi r squared, lateral_surface_area as pi r l, total_surface_area as base plus lateral which equals pi r times r plus l, 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 radius 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 segment as a real cone.
How the closed forms are evaluated: volume, slant, lateral and total surface
The engine resolves radius from the field radius with aliases r, R, and base_radius, 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 slant-height hypotenuse yourself.
Where cone metrics fit in study, design, and code
Students meet the cone when a textbook asks for volume from radius and height, when an exam demands lateral area given the 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 radii and heights, and that refuse zero or negative dimensions instead of printing NaN. Mechanical and civil design often needs the capacity of a conical hopper, the material area of a funnel wall, or the total sheet metal for a nose cone; knowing that a requested radius is not positive is as valuable as the number itself. Surveying and CAD workflows place conical 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 ellipse metrics, chord length, 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 a cone volume calculator or for slant height and surface area from radius and height, the contract is the same: positive radius and height in, finite metrics or a clear invalid_input out.
Integration use cases
Deterministic geometry fixtures
Generate reproducible cone metrics for unit tests, contest validators, and CI assertions without hand-rolled one-third volume or slant-height logic.
Technical questions
What if radius or height is zero or negative?
The handler returns invalid_input with a message that the dimension must be a positive finite number.
How is slant height computed?
Slant height l is the hypotenuse of radius and height: l = √(r² + h²). For r = 3 and h = 4 the slant height is exactly 5.
Does total surface area include the base?
Yes. total_surface_area is base_area plus lateral_surface_area, equivalently π r (r + l). lateral_surface_area alone is the side wall only.
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.