Interior and exterior triangle angles calculator
Two angles of 60° and 80° leave 40° for the third, and the exterior angles come out as 120°, 100° and 140°. Enter the two interior angles you know and get the full set at once. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle a
- 60
- Angle b
- 80
- Interior sum degrees
- 180
- Exterior sum degrees
- 360
The interior and exterior triangle angles calculator turns two known interior angles into the complete angular picture of a triangle. Every triangle's three interior angles sum to exactly 180 degrees, so the third angle is whatever remains after subtracting your two known measures. Each exterior angle is the supplement of its adjacent interior angle — the two sit on a straight line and add to 180 degrees — and the three exterior angles always total 360 degrees. You send two interior angles in degrees, each strictly between 0 and 180 and summing to less than 180, and you receive all three interior angles, all three exterior angles, and both sums.
Interior and exterior angles of a triangle
An interior angle is the angle formed inside the triangle at each vertex, between the two sides that meet there. The three interior angles always add up to 180 degrees, a fact that follows from drawing a line through one vertex parallel to the opposite side and chasing alternate angles. An exterior angle is formed at a vertex when one side of the triangle is extended beyond it; it sits on a straight line with its adjacent interior angle, so the two are supplementary and sum to 180 degrees.
A useful corollary, the exterior angle theorem, states that an exterior angle equals the sum of the two remote interior angles — the exterior angle at the third vertex is exactly the sum of your two known angles. The three exterior angles, one at each vertex, always total 360 degrees, the measure of one full turn.
The interior and exterior triangle angles calculator applies these relationships directly: given any two interior angles, it recovers the third by subtraction from 180, then derives every exterior angle as a supplement, returning the complete set in one response.
Where it fits in practice
The natural home is geometry coursework — checking homework, generating exercises, verifying student answers — but triangle angles surface anywhere a three-sided figure is drawn or measured. Roof framers checking gable pitches, surveyors closing a triangular traverse, drafters laying out a truss, game developers orienting a sprite along a polygon edge: all of them, sooner or later, complete a triangle from two known angles or convert an interior angle into the turn you make at a vertex, which is precisely the exterior angle.
What you can do with it
Complete a triangle in homework
Enter the two interior angles given in the problem and instantly get the third angle plus every exterior angle, ready to check against your working.
Compute turn angles at each vertex
The exterior angle is the turn a path makes at a corner; get all three turn angles of a triangular route or polygon edge from two known corners.
FAQ
What are the interior and exterior angles of a triangle?
The interior angles are the three angles inside the triangle at its vertices; they always sum to 180 degrees. An exterior angle is formed at a vertex when a side is extended, and it equals 180 degrees minus the adjacent interior angle.
Which inputs are accepted?
Two finite numbers, each strictly between 0 and 180 degrees, whose sum is strictly below 180.
Why is an error returned when the two angles sum to 180 or more?
Because the third interior angle would be zero or negative, so no triangle can have those angles.
Do the exterior angles always sum to 360 degrees?
Yes — one exterior angle at each vertex always totals 360 degrees, and the response includes both the interior sum of 180 and the exterior sum of 360 for verification.
Does it work with decimal angles?
Yes. Measures like 45.5 or 72.25 are fine, and every result is rounded to six decimal places so answers 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/interior-and-exterior-triangle-angles \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle_a":60,"angle_b":80}'const res = await fetch("https://api.kit.forhosting.com/math/interior-and-exterior-triangle-angles", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle_a": 60,
"angle_b": 80
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/interior-and-exterior-triangle-angles",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle_a": 60,
"angle_b": 80
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/interior-and-exterior-triangle-angles", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle_a":60,"angle_b":80}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle_a":60,"angle_b":80}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/interior-and-exterior-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
{
"angle_a": 60,
"angle_b": 80
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.interior_and_exterior_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.
How the computation works
The input is a pair of numbers: the measures of two interior angles in degrees. When the input is valid, the third interior angle is computed as 180 minus the sum, each exterior angle as 180 minus its adjacent interior angle, and every result is rounded to six decimal places. The endpoint validates that both are present, that each parses as a finite number, and that each lies strictly between 0 and 180 — an interior angle cannot be zero, negative, straight or reflex. It then checks the decisive condition: the sum of the two angles must be strictly below 180 degrees, because otherwise nothing positive remains for the third angle and no such triangle exists. When the sum reaches or exceeds 180, the request is rejected with an invalid_input error that reports the offending sum, instead of returning a zero or negative angle dressed up as an answer. That rounding is deliberate: raw floating point subtraction 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
Having it as a tiny deterministic endpoint means a spreadsheet, a tutoring app or a build script can get the full angular set — all three interior angles, all three exterior angles, and both sums — without embedding its own validation rules, and can rely on the same strictness about degenerate input everywhere it runs. 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 or drafting tool
Generate triangle exercises or validate drafted figures against a deterministic endpoint that rejects impossible angle pairs consistently.
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.