Isosceles triangle angles
Enter the vertex angle of an isosceles triangle and get its two equal base angles: a 40° vertex leaves 140° to share, so each base angle is 70°. Any vertex angle strictly between 0 and 180 degrees works. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Vertex angle
- 40
- Vertex angle degrees
- 40
- Base angle degrees
- 70
- Base angles degrees
- 70, 70
The isosceles triangle angles calculator solves one of the most common small problems in geometry: you know the vertex angle of an isosceles triangle — the angle between the two equal sides — and you need the two equal base angles. Because the interior angles of any triangle add up to 180 degrees and the base angles of an isosceles triangle are equal, each base angle is simply half of 180 minus the vertex angle. You send the vertex angle in degrees — anything strictly greater than 0 and strictly less than 180 — and you get back both base angles, also in degrees.
What makes a triangle isosceles
An isosceles triangle is a triangle with two sides of equal length. Those equal sides meet at the apex, and the angle between them is called the vertex angle; the third side, opposite the apex, is the base, and the two angles at its ends are the base angles. The key theorem — known since Euclid and sometimes called the pons asinorum — says the angles opposite the equal sides are themselves equal, so the two base angles always have the same measure.
Combined with the fact that the three interior angles of any triangle sum to 180 degrees, this means the whole angle structure of an isosceles triangle is fixed by a single number: once you know the vertex angle, the base angles are determined. The isosceles triangle angles calculator applies exactly that reasoning. A vertex angle of 40 degrees leaves 140 degrees to split evenly, so each base angle is 70 degrees; a vertex of 90 degrees — the isosceles right triangle — gives two base angles of 45 degrees each.
How the computation works
The input is a single number: the measure of the vertex angle in degrees. Both bounds are exclusive, because at exactly 0 the two equal sides would lie on top of each other and the triangle would collapse into a line segment, and at exactly 180 the base angles would both be zero — neither case is a triangle in any useful sense.
The computation itself is one line of arithmetic: subtract the vertex angle from 180 to get the combined measure of the two base angles, then divide by two to get each one.
Where it fits in practice
Isosceles triangles appear constantly outside the classroom. Roof trusses and A-frame structures are isosceles by design, so a carpenter who knows the pitch at the ridge can get the angles at the eaves. Gable ends, pediments, isosceles trapezoid layouts, symmetric road profiles and pie-slice diagrams all reduce to the same one-number problem.
Having it as a tiny deterministic service means a spreadsheet, a CAD script or a homework checker can get the answer without embedding its own arithmetic and its own validation rules, and can rely on the same strictness about the 0-to-180 range everywhere it runs.
What you can do with it
Check geometry homework
Verify that a triangle with a given vertex angle really is consistent with the base angles a student computed.
Design a roof or gable
Get the eave angles of a symmetric truss from the ridge angle, in a script or a spreadsheet formula.
FAQ
What is the vertex angle of an isosceles triangle?
The angle between the two equal sides, at the apex. The two angles at the ends of the base — the base angles — are always equal to each other.
How are the base angles computed?
Each base angle equals (180 − vertex) ÷ 2, because the interior angles of a triangle sum to 180 degrees and the two base angles are equal.
Which inputs are accepted?
Any finite number strictly between 0 and 180, interpreted as degrees.
Why are 0 and 180 degrees rejected?
Because neither describes a real triangle: at 0 degrees the equal sides collapse onto each other, and at 180 degrees both base angles would be zero.
Does it work with decimal measures?
Yes. Vertex angles like 33.7 or 102.25 are fine, and the base angles are rounded to six decimal places so results are stable across machines.
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/isosceles-triangle-angles \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"vertex_angle":40}'const res = await fetch("https://api.kit.forhosting.com/math/isosceles-triangle-angles", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"vertex_angle": 40
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/isosceles-triangle-angles",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"vertex_angle": 40
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/isosceles-triangle-angles", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"vertex_angle":40}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"vertex_angle":40}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/isosceles-triangle-angles", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"vertex_angle": 40
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.isosceles_triangle_angles",
"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
The same deterministic code runs free in your browser on this page and through the paid API, so the answer you see here is exactly the answer your integration receives.
What makes a triangle isosceles
Any vertex measure strictly between 0 and 180 degrees is valid, and the endpoint rejects anything outside that open interval with a clear reason instead of a meaningless result.
How the computation works
The endpoint validates that it is present, that it parses as a finite number, and that it lies strictly between 0 and 180. The result is rounded to six decimal places, and the response echoes your vertex angle back alongside the base angle measure and a two-element array with both base angles, so a log, a report or an audit trail can show the complete angle picture without a second call. The rounding is deliberate: raw floating point arithmetic can differ in the last bit between machines, and a stable, rounded result guarantees that the free web version and the paid API return byte-identical answers every time.
Where it fits in practice
In graphics and games, symmetric shapes and mirrored splines lean on the same identity, and a tutoring app can generate endless exercises from a single endpoint. It costs $0.002 per request through the API, with no network calls or hidden state on our side, and the identical logic runs free in the browser on this page for one-off checks before you wire it into anything.
Integration use cases
Power a tutoring app
Generate isosceles-triangle exercises and validate student answers against a deterministic, always-consistent endpoint.
Technical questions
What does it cost?
$0.002 per request through the API. It is also free to run in your browser on this page, with the same code and the same answer.