Right circular cone
A cone with radius 3 and height 4 has a slant height of exactly 5 and a volume of 37.6991118431 cubic units. Enter the radius and height to get the volume, the lateral area and the total surface including the base. 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
- Lateral area
- 47.1238898038
- Total surface area
- 75.3982236862
A right circular cone is the solid formed when a right triangle rotates about one of its legs: the other leg becomes the base radius, the axis of rotation is the perpendicular height, and the hypotenuse traces the slant height that generates the lateral surface. This calculator accepts those two positive lengths — radius and height — and returns the three measurements that homework, design, and code most often need: the enclosed volume, the lateral surface area of the unrolled sector, and the total surface area that adds the circular base.
What a right circular cone is and which three measurements this tool returns
A right circular cone is completely determined by two positive lengths: the radius of the circular base and the perpendicular height from the apex down to the center of that base. The axis that joins the apex to the center is orthogonal to the base plane; that is what “right” means, as opposed to an oblique cone whose apex sits off-center.
Rotating a right triangle about the height leg produces the solid: the other leg sweeps the disk of radius r, and the hypotenuse sweeps a sector whose radius is the slant height l equal to the square root of r squared plus h squared. This calculator does not ask you to supply that slant height.
It derives l internally and then publishes the three measurements people actually look up: volume V equals one third pi r squared h, which is one third of the cylinder that shares the same base and height; lateral area A_lateral equals pi r l, the area of the unrolled sector; and total surface area A_total equals pi r times the sum of r and l, which is simply the lateral area plus the circular base. Each must be finite and strictly greater than zero.
Formula strings travel with the numbers so a student or a log line can verify the identities without opening the source. If you need the volume of an oblique cone, a frustum, or a cylinder, use a tool dedicated to those solids; the search intent here is the right circular cone from radius and height, with a clear answer or an honest error when a dimension is not positive.
How the closed forms are evaluated from radius and height
After parsing, the engine insists that each dimension is strictly greater than zero. Zero is not a cone, a negative length is not a geometric measure, and Infinity or NaN cannot produce a finite area, so those cases fail before any formula runs. The classic classroom check is radius three and height four, which yields slant height exactly five, the 3-4-5 Pythagorean triple.
Volume is one third times pi times radius squared times height, independent of any series expansion or sampling. Lateral area multiplies pi, radius, and that slant height, which is the area you would obtain by unrolling the side into a circular sector of radius l and arc length 2 pi r. Total surface area adds the circular base pi r squared to that lateral area, equivalently pi r times the quantity r plus l.
Where right circular cone metrics fit in study, design, and code
Students meet the right circular cone when a textbook asks for volume from radius and height, when an exam demands lateral area after the slant height has been found, or when solid-geometry homework wants total surface including the base before a paint or material estimate. Teachers can pre-generate answer keys that list volume, lateral area, and total surface 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 sheet-metal area of a funnel wall, or the total wrap of 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.
Remember the contract limits: this tool does not size a frustum, an oblique cone, or a cone packing problem, and it does not convert units. If radius is in centimeters and height in centimeters, volume is in cubic centimeters and areas in square centimeters.
What you can do with it
Homework and exam right-circular-cone drills
Compute volume, lateral area, 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 perpendicular height in CAD and manufacturing tools.
FAQ
What does the right circular cone calculator cost?
It is free in your browser on this page.
What if radius or height is zero or negative?
A degenerate segment is not a right circular cone.
How are volume, lateral area, and total surface computed?
Volume is (1/3)πr²h. Slant height l is √(r²+h²). Lateral area is πrl. Total surface is πr(r+l), which is the lateral area plus the circular base.
Do I need to supply the slant height?
No. The engine derives slant height from radius and height using the Pythagorean theorem and then uses that length for the two surface formulas.
Is this the same as an oblique cone or a frustum?
No. This tool is only for a right circular cone from positive radius and perpendicular height. Oblique cones and frustums need their own contracts.
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/math/right-circular-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/math/right-circular-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/math/right-circular-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/math/right-circular-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/math/right-circular-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": "math.right_circular_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
Radius and height must be finite and strictly greater than zero; a missing, zero, negative, or non-numeric value fails with invalid_input instead of a silent NaN. Through the API a successful request costs $0.002; validation failures are not billed.
What a right circular cone is and which three measurements this tool returns
Preferred inputs are the JSON numbers radius and height. Missing fields, empty strings, NaN, Infinity, zero, and negative values all raise invalid_input with a message that names the failing field. The response echoes the validated radius and height, reports the derived slant height so you can check the Pythagorean step, and then lists volume, lateral_area, and total_surface_area.
How the closed forms are evaluated from radius and height
The engine reads two required fields from the JSON body: radius and height. Each value may arrive as a JSON number or as a numeric string; both are parsed to a finite IEEE double. Once both dimensions pass, slant height is the Euclidean hypotenuse of radius and height: the square root of r squared plus h squared. 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 right circular cone metrics fit in study, design, and code
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 — lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Whether you arrived searching for a right circular cone calculator or for volume, lateral area, and total surface from radius and height, the contract is the same: positive radius and height in, three finite measurements or a clear invalid_input out.
Integration use cases
Deterministic geometry fixtures
Generate reproducible right circular cone metrics for unit tests, contest validators, and CI assertions without hand-rolled one-third volume or slant-height logic.