Triangle area from two sides and included angle calculator
Sides of 3 and 4 with a 90° angle between them enclose an area of 6. Enter two side lengths in one unit and the included angle in degrees to get the triangle's area in the matching square unit. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Side a
- 3
- Side b
- 4
- Included angle degrees
- 90
- Area
- 6
This triangle area calculator finds the area when you know two side lengths and the angle between them. It applies the side-angle-side area formula, one half times side a times side b times the sine of the included angle. Enter both sides in the same length unit and provide the angle in degrees. The result is expressed in the corresponding square unit. Positive-side validation and a strict angle range prevent collapsed or invalid triangles, while deterministic arithmetic makes the result suitable for homework, design checks, and automated calculations.
Identify the included angle before calculating
The formula works with two sides and the angle physically enclosed by those sides. If the known lengths are side a and side b, the required angle is the angle at the vertex where a and b meet. This detail is essential: an angle opposite either known side is not the included angle and cannot simply be substituted into the same expression.
A length of three meters and a length of four meters with a ninety-degree included angle are valid inputs. Mixing three meters with four feet is not meaningful unless one value is converted first. The calculator does not attach a physical unit because the arithmetic is valid for any consistent system.
If the sides are centimeters, the returned area represents square centimeters; if they are feet, it represents square feet. A simple sketch with the known angle marked between the known edges is often the quickest way to confirm that the correct angle has been selected before submitting the calculation.
Understand the half-ab-sine formula
Triangle area can be written as one half times a base times its perpendicular height. When side a is treated as the base, the perpendicular component of side b is b times sine C, where C is the included angle. Substituting that height gives A = one half a b sine C.
The calculator converts the supplied degree measure to radians for the JavaScript sine function, evaluates the expression, and returns a stable numeric result. For a three-by-four triangle with a ninety-degree included angle, sine ninety degrees is one, so the area is one half times three times four, or six square units. Acute and obtuse included angles are both supported.
Angles such as sixty and one hundred twenty degrees have the same sine, so the same pair of sides produces the same area even though the triangles have different shapes. Angles equal to zero or one hundred eighty degrees would put all vertices on a line and produce zero area; they are rejected rather than reported as valid triangles. This method requires no third side, altitude, perimeter, or iterative solver.
What you can do with it
Check trigonometry homework
Verify an area calculation when a problem provides two sides and the angle between them.
Estimate a triangular panel
Calculate surface area from two measured edges and their included corner angle before ordering material.
Automate survey calculations
Turn consistent side and angle measurements into reproducible square-unit areas in a script or data pipeline.
FAQ
Which angle should I enter?
Enter the angle between the two known sides, at the vertex where those sides meet.
What formula does the calculator use?
It uses A = ½ab sin(C), where a and b are the known sides and C is their included angle.
What units does the result use?
The result uses square versions of the side unit. Sides in meters produce square meters; sides in inches produce square inches.
Why do 60° and 120° give the same area?
Both angles have the same sine, so with the same two sides the area comes out equal even though the triangles have different shapes. The formula depends on the sine of the included angle.
Can the included angle be 0 or 180 degrees?
No. At 0 or 180 degrees all three corners would sit on one line and the area would be zero, so the calculator asks for an angle strictly between 0 and 180 instead.
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/trig/area-sas \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"side_a":3,"side_b":4,"included_angle_degrees":90}'const res = await fetch("https://api.kit.forhosting.com/trig/area-sas", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"side_a": 3,
"side_b": 4,
"included_angle_degrees": 90
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/area-sas",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"side_a": 3,
"side_b": 4,
"included_angle_degrees": 90
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/area-sas", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"side_a":3,"side_b":4,"included_angle_degrees":90}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"side_a":3,"side_b":4,"included_angle_degrees":90}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/area-sas", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"side_a": 3,
"side_b": 4,
"included_angle_degrees": 90
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.area_sas",
"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
Identify the included angle before calculating
Enter side_a and side_b as positive numbers measured in one consistent unit, then enter included_angle_degrees as a value strictly between zero and one hundred eighty.
Validate, interpret, and automate the result
Both side fields must be finite numbers greater than zero, and the angle must be a finite degree value inside the open interval from zero to one hundred eighty. On success, the response includes the computed area, the three normalized inputs, and the formula label used for the calculation. Keeping the inputs beside the answer is useful in logs, spreadsheets, grading systems, and engineering scripts because a reviewer can see exactly which dimensions produced the result. The operation is pure and deterministic: it uses no network request, current date, random value, or stored state. Missing values, text values, infinities, negative lengths, zero lengths, and endpoint angles return an invalid_input error. These checks help prevent a plausible-looking number from masking an invalid geometric model. Repeating the same request therefore produces the same JSON. For batch work, convert every pair of sides to a shared unit before calling the API and preserve that unit in your own record. Use $0.002 when invoking the API, or run the same calculation in the browser for an immediate manual check.
Technical questions
Can the angle be zero or one hundred eighty degrees?
No. Those endpoint angles collapse the triangle to a line and produce zero area, so the calculator rejects them.
How much does an API calculation cost?
Each API request costs $0.002. The browser calculation is available without an API request.