Hemisphere calculator
Enter one radius and get a hemisphere's volume, curved dome area and total surface together: radius 1 gives a volume of two thirds of π, a dome of 2π and a total of 3π. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 5
- Volume
- 261.7993877991
- Curved surface area
- 157.0796326795
- Total surface area
- 235.6194490192
A hemisphere is half of a solid sphere cut by a plane through its center, leaving a curved dome and a flat circular base. Given a single positive radius, this calculator returns the enclosed volume, the curved surface area of the dome, and the total surface area that adds the base disk.
How volume, curved and total surface area are computed
Label the distance from the center of the full sphere to any point on the curved surface as the radius r. The volume of a solid hemisphere is two-thirds pi times r cubed: volume equals (2/3) π r³. That is exactly half the classic sphere volume (4/3) π r³ and measures the cubic capacity inside the dome and down to the flat cut.
The curved surface area is the lateral area of the dome alone, equal to 2 π r²—half of the full sphere surface 4 π r²—and does not include the base. Classic checks include radius one, which yields volume about 2.094395102393, curved surface about 6.28318530718, and total surface about 9.424777960769; radius three yields volume about 56.548667764616 and total surface about 84.823001646924; radius five is a common textbook case with volume about 261.799387799149.
Because the path uses only multiplication, a fixed fraction two-thirds, and Math.PI, it is deterministic, O(1), and free of iterative solvers.
Where hemisphere calculations show up in real work
Secondary-school and college solid-geometry courses constantly ask students for the volume and surface areas of a hemisphere given only the radius, and often distinguish curved area from total area on the same problem set; a trusted calculator with a strict positive-radius gate speeds homework grading and removes spreadsheet mistakes where someone uses the full-sphere volume or forgets the base in total surface.
Architecture, roofing and fabrication workflows size geodesic and masonry domes, planetarium shells and decorative half-spheres from outer or design radius; volume feeds capacity and concrete estimates while curved area informs cladding and paint and total area includes the footprint or base seal. Storage tanks, silos and chemical vessels are sometimes modeled as a cylinder with hemispherical ends; the hemisphere half is reused for end-cap volume and shell area before the cylindrical body is added.
Game, graphics and CAD libraries store half-sphere colliders and primitives; deterministic volume and surface metrics support debug readouts and material estimates without a full mesh library. If you later need a full sphere, a spherical cap of arbitrary height, or a hollow thin shell, treat those as separate search intents rather than overloading this hemisphere landing.
What you can do with it
Homework and exam checks
Compute hemisphere volume, curved surface area and total surface area from radius, and reject non-positive radius with a clear error.
Domes, tanks and fabrication
From design radius, recover capacity volume, cladding curved area and total surface including the base footprint.
CAD and game half-sphere metrics
Turn a single radius into volume and surface figures for layout, material estimates and property panels.
FAQ
What happens if the radius is zero or negative?
The radius must be positive (strictly greater than zero); the error message names the field.
Is this the same as a full sphere?
No. A full sphere has volume (4/3) π r³ and surface 4 π r². This capability is the solid hemisphere only—half the volume and half the curved area, plus the base for total surface.
What does it cost?
The browser tool is free and runs locally.
What is the difference between curved and total surface area?
The curved area covers the dome alone and equals 2πr², half the surface of a full sphere. The total adds the flat circular base of πr², so it comes to 3πr².
What is the formula for the volume of a hemisphere?
Two thirds of π times the radius cubed, exactly half the volume of the full sphere. It measures the capacity inside the dome, all the way down to the flat cut.
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/hemisphere \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":5}'const res = await fetch("https://api.kit.forhosting.com/geom/hemisphere", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/hemisphere",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/hemisphere", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/hemisphere", 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": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.hemisphere",
"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
A non-positive radius 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 radius you provide.
How volume, curved and total surface area are computed
The total surface area adds the flat circular base of area π r², so total_surface_area equals 3 π r². Huge magnitudes can still overflow IEEE-754 double range; when intermediates are 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 radius.
What you send and what you get back
Provide one required finite number: radius is the radius of the hemisphere and must be strictly greater than zero. Zero and negative values are rejected; there is no optional height for a spherical cap—this landing is the solid half-sphere only. Example checks: radius one yields volume two-thirds pi, curved surface two pi, and total surface three pi; radius five scales by r cubed for volume and r squared for both areas; a zero or negative radius fails naming the field and the positivity rule. Wire the free browser tool for quick sanity checks, then call the identical contract from scripts for bulk evaluation without reimplementing the hemisphere formulas in every language. Missing fields, empty strings, non-numeric text, NaN, Infinity, and any non-positive radius are rejected with invalid_input and are not charged. On success the JSON response returns radius, volume, curved_surface_area, total_surface_area, and method set to hemisphere_volume_curved_total_surface so logs and teaching UIs can document the algorithm. API pricing is the published base of $0.002 per successful item.
Where hemisphere 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?
volume = (2/3) π r³; curved_surface_area = 2 π r²; total_surface_area = 3 π r² (curved dome plus flat base π r²). The response includes method: "hemisphere_volume_curved_total_surface".
Does total surface area include the flat base?
Yes. curved_surface_area is the dome only (2 π r²); total_surface_area adds the circular base (π r²) for 3 π r² overall.