Cone apex angle calculator
Get the full opening angle of a cone, twice arctan(r/h), from its base radius and its height; radius 3 and height 4 give about 73.7397952917°. The half-angle comes too, in degrees and radians. 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
- Half apex angle rad
- 0.6435011088
- Half apex angle deg
- 36.8698976458
- Apex angle rad
- 1.2870022176
- Apex angle deg
- 73.7397952917
The apex angle of a right circular cone is the plane angle at the tip of an axial cross-section: cut the cone through its axis and you obtain an isosceles triangle whose equal sides are generatrices and whose base is a diameter of the circular base. That triangle’s vertex angle is the full apex angle—twice the half-angle from the axis down to a generatrix. When drawings, optics notes, packing specs, or homework already give the base radius and the perpendicular height, the missing step is a single inverse-tangent evaluation: half-angle alpha equals arctan of radius over height, and the full apex angle is twice that value.
What the full apex angle means and why θ = 2 arctan(r/h)
In solid geometry a right circular cone is determined by a base radius r and a perpendicular height h measured from the apex to the center of the base disk. An axial cross-section through the axis and a diameter of the base yields an isosceles triangle of base two r and height h.
From the apex, drop a perpendicular to the base center: each right half of that triangle has opposite side r and adjacent side h, so the half apex angle alpha between the axis and a generatrix is arctan of r over h. The full apex angle theta—the opening angle people mean when they say cone apex angle, cone opening angle, or vertex angle of a cone—is twice alpha, which is exactly two times arctan of r over h.
Search intent clusters around phrases such as cone apex angle calculator, full apex angle from radius and height, how to find cone vertex angle, and opening angle of a cone. People want a numeric answer they can check against a textbook key, an optics aperture note, or a manufacturing drawing, not only a reminder that the angle sits in a right triangle. Units stay consistent: radius and height must share a length unit; the engine does not convert units and does not invent a preferred unit system.
The formula is independent of the absolute scale of length units because only the ratio r over h enters the inverse tangent.
What to enter and what you get back
Zero height is always rejected with a clear message that a cone with zero height has no defined apex angle, matching the product contract and the mathematical fact that arctan of r over h would divide by height. Non-positive radius and non-positive height other than the explicit zero case are likewise rejected before any angle is returned. Worked example: radius three and height four give half-angle arctan of three fourths and full apex angle twice that value in both radians and degrees.
Another check: equal radius and height yield a half-angle of forty-five degrees and a full apex angle of ninety degrees. Edge cases that fail before any metric is returned include height zero, negative height, non-positive radius, non-numeric strings, infinity, and missing fields. Failed validation is not billed.
Practical uses, limits, pricing, and delivery channels
Use the free browser tool when you want an immediate full apex angle from known radius and height while checking solid-geometry homework, verifying a conical packing tip angle, reading an optical cone opening from a drawing, or confirming a workshop sketch without installing a computer-algebra system. Results are fully deterministic: the same radius and height always yield the same half and full apex angles. There is no language model, no external geometry service, and no randomness, so the capacity is safe to pin in continuous-integration checks and reproducible lesson materials.
Limits are intentional: the tool does not convert among unit systems, does not solve oblique cones or frustums, and does not emit diagrams—only the closed-form full apex angle of a right circular cone from its radius and height.
What you can do with it
Homework and solid-geometry drills
Students and graders confirm the full cone apex angle from radius and height, including clear rejection when height is zero or either length is not positive.
FAQ
What happens if height is zero?
A cone with zero height has no defined apex angle because the ratio r/h is undefined; validation failures are not billed.
Does this compute cone volume or surface area?
No. This capacity is apex angle only. Use a cone volume calculator when you need volume, slant height, or surface metrics from radius and height.
How do I calculate the full apex angle of a cone?
Divide the base radius by the height, take the arctangent to get the half-angle, then double it. With a radius of 3 and a height of 4 the half-angle is 36.8698976458° and the full angle 73.7397952917°.
What is the difference between the half-angle and the full apex angle?
The half-angle runs from the axis of the cone to its sloping side. The full apex angle is twice that, the angle at the tip of the triangle you see when you cut the cone through its axis.
Does the unit of length matter?
No, only the ratio of radius to height counts, so doubling both leaves the angle unchanged. Just use the same unit for both, since the calculator does not convert units.
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-apex-angle \
-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-apex-angle", {
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-apex-angle",
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-apex-angle", 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-apex-angle", 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_apex_angle",
"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
This cone apex angle calculator accepts positive finite radius and height, rejects zero height with invalid_input, and returns half and full angles in both radians and degrees as structured JSON.
What the full apex angle means and why θ = 2 arctan(r/h)
Returning radius, height, half angles, and full angles in radians and degrees as JSON makes the result easy to paste into notes, pin in automated tests, or feed into a larger pipeline without reimplementing inverse tangents in every language you teach or deploy.
Input contract, formulas, validation, and response shape
POST a JSON body with the required fields radius and height, each a positive finite number. Either field may arrive as a JSON number or as a numeric string; aliases r, R, base_radius for radius and h, H for height are accepted. Missing, empty, or non-finite values fail with invalid_input naming the field. On success the engine computes half_apex_angle_rad as Math.atan of radius over height, apex_angle_rad as twice that value, converts both to degrees with the usual one-hundred-eighty over pi factor, and rounds derived quantities to a stable decimal precision so the same input always yields the same JSON across JavaScript engines. The response echoes radius and height, exposes half_apex_angle_rad, half_apex_angle_deg, apex_angle_rad, apex_angle_deg, a short formula string restating theta equals two arctan of r over h, and method set to cone_apex_angle_from_radius_height. This endpoint does not return volume or surface area; those belong to a different SEO landing and a different capacity.
Practical uses, limits, pricing, and delivery channels
Each successful request is one billable item at $0.002; failed validation does not consume balance. Accepted inputs must be strictly positive; zero height is always rejected with invalid_input. Wire the endpoint into solid-geometry drills, contest answer checkers, manufacturing worksheets that list only radius and height, or floating preprocessing steps where you already prefer explicit domain errors over opaque black-box widgets that cannot show why zero height is rejected.
Integration use cases
Manufacturing and packing tip angles
Shop drawings that list base radius and height recover the opening angle as structured JSON for tooling, nozzles, or conical pack tips.
Optics and cone aperture checks
Notebooks that need the full opening angle of a right circular cone from measured r and h call a deterministic parse API instead of reimplementing arctan in every language.
Technical questions
How do I calculate the full apex angle of a cone?
Send positive finite radius and height. The API returns half-angle α = arctan(r/h) and full apex angle θ = 2α in both radians and degrees.
Is the apex angle the half-angle or the full opening?
This calculator returns both. The primary product result is the full apex angle θ = 2 arctan(r/h); half_apex_angle_* fields expose α for axis-to-generatrix work.
What does the cone apex angle calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.