Similar right triangles
If side 3 of a right triangle with sides 3 and 4 becomes 6 on a similar copy, the 4 becomes 8: the scale factor is 2. Enter any three of the four corresponding sides and get the missing one and the scale. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 3
- A prime
- 6
- B
- 4
- Unknown side
- 8
- Unknown
- b_prime
- B prime
- 8
- Scale factor
- 2
Two right triangles are similar when their corresponding acute angles match, so every pair of corresponding sides shares the same scale factor. Geometry homework, drafting layouts, and roof or ramp sketches routinely give three of those four lengths — a side on the first triangle, its partner on the second, and one more corresponding side — and ask for the missing length. This similar right triangles calculator applies the closed-form proportion of corresponding sides: the unknown equals the known partner times the ratio of the other corresponding pair.
What similar right triangles share and how the unknown side appears
Two Euclidean right triangles are similar when one acute angle of the first equals one acute angle of the second. The right angles already coincide, so AA similarity follows from a single shared acute angle, and every pair of corresponding sides then stands in one common scale factor. Given any three of those four positive lengths, the fourth is a single multiplication and division: the missing side equals its known partner times the ratio of the other corresponding pair.
That closed form is not a search, a numeric fit, or an appeal to the Pythagorean theorem — the triangles are already known to be similar, so the scale factor alone recovers the unknown. Classic classroom triples appear immediately: sides three and four on a three-four-five triangle correspond to six and eight on a six-eight-ten copy, so three known members of that corresponding quartet recover the fourth as eight, six, four, or three. A scale of three halves sends six and eight to nine and twelve.
Because a Euclidean length cannot be zero or negative, every given side must be a positive finite number; a zero run, a signed cut, Infinity, or a non-numeric token is not a geometric side and must not look like a successful missing-side solve. This calculator encodes that contract end to end and returns the unknown length together with the completed corresponding quartet and the scale factor from the first triangle to the second.
Where similar right triangles come in handy
Classrooms and tutoring platforms use a similar right triangles calculator so learners can verify the missing corresponding side without mixing up which ratio to write first. Construction and layout tools convert a measured pair on a small template plus one measured member on the full-size copy into the remaining cut length before ordering stock. Roof, ramp, and stair sketches often copy a right triangle at a new scale: three known corresponding sides fix the fourth without reopening a trigonometry table.
Game engines and physics sandboxes spawn a similar copy from whichever three corresponding lengths the designer already stored. Exam generators and LMS plugins need a deterministic oracle that rejects a non-positive side instead of taking absolute values and marking a signed homework token as correct. The landing is deliberately narrower than a full right-triangle solver: it does not return angles or area and it does not invent a hypotenuse from two legs, so the page stays aligned with the search for a missing corresponding side of two similar right triangles.
What you can do with it
Classroom corresponding-side checks
Students enter any three corresponding sides of two similar right triangles to verify the missing length against textbook 3-4-5 and 6-8-10 similarity exercises.
Scaled template to full-size cut
Convert a measured pair on a small right-triangle template plus one measured member on the full-size copy into the remaining cut length before ordering material.
Exam and LMS auto-grading
Mark similar-right-triangle items with a deterministic oracle that rejects a non-positive given side and never invents a fourth length from the wrong pair.
FAQ
What formula does the similar right triangles calculator use?
Corresponding sides of similar right triangles are proportional. The unknown equals its known partner times the ratio of the other corresponding pair.
What happens if a given side is zero or negative?
Every given side must be a positive finite number; those validation failures are not billed.
Which three sides do I need to enter?
Two sides of the first triangle and the partner of one of them on the second, or any other three of the four corresponding sides. Leave the unknown one out; entering only two or all four is refused.
What does the scale factor tell me?
How many times bigger the second triangle is than the first. With 3 becoming 6 the factor is 2; a factor of three halves would turn sides 6 and 8 into 9 and 12.
Do I need an angle or the Pythagorean theorem?
No. The triangles are taken as similar, which already holds when they share one acute angle, so the scale between corresponding sides is enough to find the missing length.
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/similar-right-triangles \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":3,"side_a":1}'const res = await fetch("https://api.kit.forhosting.com/math/similar-right-triangles", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 3,
"side_a": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/math/similar-right-triangles",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 3,
"side_a": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/math/similar-right-triangles", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":3,"side_a":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":3,"side_a":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/math/similar-right-triangles", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 3,
"side_a": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "math.similar_right_triangles",
"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
Every given side must be a positive finite number; zero, negative, infinite, and non-numeric values fail with invalid_input and are not billed.
What similar right triangles share and how the unknown side appears
Label a side of the first triangle a and its partner on the second triangle a_prime; label another corresponding pair b and b_prime. The defining proportion is a over a_prime equals b over b_prime, which is the same statement as a times b_prime equals a_prime times b.
Input contract, response shape, and error behaviour
POST a JSON body with exactly three of the four corresponding sides: a and b belong to the first right triangle, and a_prime and b_prime are their partners on the second similar right triangle. Each field may arrive as a JSON number or as a numeric string. Field a also accepts the aliases side_a and first_a; a_prime accepts a2, corresponding_a, and ap; b accepts side_b and first_b; and b_prime accepts b2, corresponding_b, and bp. Missing the trio, sending only one or two sides, or sending all four sides fails with invalid_input asking for exactly three known corresponding sides. A given side that is zero, negative, infinite, NaN, or not a decimal number fails with invalid_input naming the field and stating that a positive finite number is required. Those validation failures are not billed. On success the body returns unknown_side as the recovered length, names the unknown field in unknown as a, a_prime, b, or b_prime, echoes the completed quartet, reports scale_factor as a_prime divided by a, restates the identity in the formula field, and sets method to similar_right_triangles so clients can distinguish this landing from a Pythagorean missing-side solver on a single triangle. Example: a equals three, a_prime equals six, and b equals four yields unknown_side eight with unknown b_prime and scale_factor two. Example: a equals six, b equals eight, and b_prime equals twelve yields unknown_side nine with unknown a_prime. Pair the structured JSON with your own drawing code if you need a sketch; this endpoint returns numbers only and does not emit SVG, step-by-step prose beyond the field names, or unit conversion — you supply the three known lengths already in one consistent unit.
Where a similar-right-triangles API earns its keep
Keep requests small, pin the rounded unknown_side in tests, and you get reproducible geometry for homework, edge workers, and design scripts without standing up symbolic math infrastructure of your own.
Technical questions
How do I find a missing side of similar right triangles?
Send exactly three of a, a_prime, b, and b_prime as positive finite numbers. The API returns the fourth length from the proportion a/a_prime = b/b_prime.
Do I have to send the hypotenuse?
No. a and b can be any corresponding pair — two legs, a leg and the hypotenuse, or any other matching sides — as long as a corresponds to a_prime and b corresponds to b_prime.
What does the similar right triangles calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.