Truncated cone calculator
A truncated cone with radii of 3 and 5 and a height of 4 has a volume of (1/3)π × 4 × 49 and a total surface of 34π + 8π√20. Enter both radii and the height to size any bucket, funnel or tapered tank. Free in your browser.
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Worked example
- Radius1
- 3
- Radius2
- 5
- Height
- 4
- Volume
- 205.2507200345
- Surface area
- 219.2111859187
A truncated cone is the solid left when a plane parallel to the base cuts a right circular cone short of the apex. Three positive lengths fix the figure: the two parallel base radii and the perpendicular height between those planes. This calculator accepts those three numbers, applies the classical volume formula one-third pi times height times the sum of the squared radii and their product, and returns the total surface area of both circular bases plus the lateral band whose width is the slant height. Students use it to check homework and contest keys; engineers, packaging designers, and makers use the same pair of numbers for hoppers, buckets, lamp shades, funnels, and tapered tanks.
What a truncated cone is and how volume and surface are determined
In solid geometry a right circular truncated cone, sometimes introduced in textbooks as a conical frustum, is the portion of a cone that lies between two planes both perpendicular to the axis. The two circular faces are parallel; their radii need not be equal, and the perpendicular distance between those faces is the height used in every closed-form formula.
Volume follows from integrating circular cross-sections whose radius interpolates linearly from one base to the other, which collapses to one-third pi times height times the sum of the two squared radii and the product of the radii. Surface area is the sum of three pieces: the two base disks, each pi times its radius squared, and the lateral band obtained by unrolling the side into a circular annulus sector.
The width of that band is the slant height, the hypotenuse of the right triangle whose legs are the height and the absolute difference of the two radii. Search intent clusters around truncated cone calculator, truncated cone volume, and truncated cone surface area: people want numeric results they can check against an answer key or a shop drawing, not only a geometric argument that the solid is determined.
The three measurements and the two results
Each must be a positive finite number in any consistent unit system. Zero is not accepted: a vanishing radius would collapse the solid into a full cone, and a vanishing height would collapse it into a flat annulus, both of which sit outside this strictly positive contract. Negative lengths are rejected for the same reason a length cannot be negative in Euclidean space.
Example: radius1 equals three, radius2 equals five, and height equals four yields volume one-third pi times four times forty-nine, and surface area equal to thirty-four pi plus eight pi times the square root of twenty. Units travel with you: if the radii and height are centimetres, volume is cubic centimetres and surface area is square centimetres.
The engine does not convert units and does not emit plots or step-by-step prose beyond the formula field, so clients that need a sketch or a unit change keep that work on their side of the contract.
Where a truncated cone calculator earns its keep
Classrooms and tutoring platforms use a truncated cone calculator so learners can check both volume and surface area for arbitrary positive radii and height without dropping a factor of one-third or forgetting a base disk. Exam-prep generators need a deterministic oracle that marks wrong a non-positive length instead of silently returning NaN that a buggy grader might treat as success. Packaging and lighting designers size tapered cartons and lamp shades from measured rims without standing up a computer-algebra system for two closed-form expressions.
Failed validation does not consume balance. Keep requests small, pin the rounded volume and surface area in tests, and you get reproducible solid-geometry numbers for homework, LMS plugins, fabrication helpers, and edge workers without importing a symbolic math library of your own.
What you can do with it
Classroom and homework checks
Let students submit two base radii and height and verify both volume and total surface area against the textbook frustum formulas.
Exam and LMS auto-grading
Mark truncated-cone items with a deterministic oracle that rejects any non-positive or non-finite length and never invents a volume.
FAQ
How do I compute truncated cone volume and surface area?
Send positive finite numbers radius1, radius2, and height.
What counts as invalid input?
Every input must be a positive finite number; those failures are not billed.
Can the two radii be equal?
Yes. Equal positive radii describe a right circular cylinder: volume becomes πR²h and total surface becomes 2πR²+2πRh, which the same formulas produce automatically.
What is the slant height of a truncated cone?
It is the width of the sloping side, the hypotenuse of a right triangle whose legs are the height and the difference between the radii. With radii of 3 and 5 and a height of 4, the slant height is √20.
Does the surface area include the top and the bottom?
Yes. The total adds both circular bases to the sloping band around the side. For radii of 3 and 5 the two bases alone account for 34π, and the band adds 8π√20 to that.
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/truncated-cone \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius1":3,"radius2":5,"height":4}'const res = await fetch("https://api.kit.forhosting.com/math/truncated-cone", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius1": 3,
"radius2": 5,
"height": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/truncated-cone",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius1": 3,
"radius2": 5,
"height": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/truncated-cone", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius1":3,"radius2":5,"height":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius1":3,"radius2":5,"height":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/truncated-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
{
"radius1": 3,
"radius2": 5,
"height": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.truncated_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
Every input must be a positive finite number: zero, negative, infinite, missing, or non-numeric values fail with invalid_input and are not billed.
What a truncated cone is and how volume and surface are determined
Adding those three pieces gives the total surface that this endpoint returns. Returning volume and surface_area as structured JSON makes the result easy to paste into notes, pin in automated tests, or feed into a larger pipeline without reimplementing pi multiplications and a square root in every language you teach or deploy.
Input contract, response shape, and error behaviour
POST a JSON body with three required fields: radius1, radius2, and height. Values may arrive as JSON numbers or as numeric strings; empty strings, missing keys, NaN, and infinities fail with invalid_input naming the field. On success the body echoes the three inputs, returns volume and surface_area rounded to twelve decimal places so goldens compare byte-for-byte across JavaScript engines, and includes a formula string restating both closed forms.
Where a truncated cone calculator earns its keep
Shop and logistics notes that list top radius, bottom radius, and height recover hopper, bucket, or tank capacity and skin area as structured JSON for reports and cutting lists.
Integration use cases
Hoppers, buckets, and tapered tanks
Recover capacity and skin area from shop measurements of the two rims and the perpendicular height as structured JSON for reports or cutting lists.
Technical questions
Does surface_area include the two bases?
Yes. surface_area is the total surface: both circular bases plus the lateral band. The lateral piece uses slant height √(h²+(R−r)²).
What does the truncated cone calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.