Similar Right Triangles Calculator - Proportional Sides and Angles

A similar right triangles calculator is a geometry tool that solves the missing sides, hypotenuse, and acute angles of two right triangles that share the same three angles. When matching legs and hypotenuses are proportional, a single scale factor (or one shared side) is enough to recover the full set of measurements for the second triangle.

• • Surveyors and field measurers: Recover the height of a building, tree, or pole by setting up a smaller right triangle next to it and using the scale factor to the object you are measuring.
• • Geometry students: Check homework, prepare for tests, and explore the relationship between a 3-4-5 right triangle and a 6-8-10, 9-12-15, or 30-40-50 scaled pair without redoing the arithmetic by hand.
• • Carpenters and roof framers: Use a smaller scaled drawing or paper-folded model to predict the rafter length, run, and rise on the full-size roof before cutting lumber.
• • Engineers and architects: Scale model dimensions to real-world dimensions on architectural drawings, scale models, and stress-test fixtures that mirror right-triangle geometry.

Most users come to this tool with one right triangle they already know (a 3-4-5 in a textbook problem, the shadow-and-height pair from a survey, or the run and rise of a roof) and a second right triangle whose sides they need.

For a more general solver that handles non-right triangles, the Triangle Calculator accepts any three of sides, angles, area, or perimeter and works backward to the rest.

The tool enforces the side-proportionality rule for similar triangles and the Pythagorean theorem for right triangles, then solves the unknowns from whichever inputs you have.

• a1, b1, c1: Leg a, leg b, and hypotenuse c of the reference right triangle (triangle 1).
• a2, b2, c2: Leg a, leg b, and hypotenuse c of the second right triangle (triangle 2).
• k: Scale factor equal to a2 / a1, b2 / b1, and c2 / c1. Entered directly or recovered from one shared side.

If you only know one side of triangle 2, the tool divides it by the matching side of triangle 1 to recover k, then multiplies the rest of triangle 1 by that same k. If you already know k, the division is skipped and k is applied directly. Either way, the solved second triangle has the same three angles as triangle 1, which is what makes the pair similar in the first place.

According to MathWorld (Wolfram Research), two triangles are similar if their corresponding angles are equal and the ratios of their corresponding sides are equal, written as a2/a1 = b2/b1 = c2/c1

The Right Triangle Calculator is the natural next step if you need to fully solve a single right triangle first, since it returns the missing legs, hypotenuse, angles, area, and perimeter from any two known values.

Four ideas explain why this calculator works and when you can trust its results.

If you remember that one right angle plus one matching acute angle locks the shape, the rest of the math falls into place: the legs and hypotenuse must line up in a fixed ratio that you can read off any single side pair.

To generate a clean integer family like (3, 4, 5), (6, 8, 10), or (30, 40, 50) for a similar right triangle, the Pythagorean Triples Calculator lists scaled Pythagorean triples whose sides are all proportional.

Walk through these steps with any pair of right triangles known to be similar, whether the scale factor is given or recovered.

1. 1 Enter the legs of triangle 1: Type the two legs of your reference right triangle. If you also know the hypotenuse, enter it; otherwise the tool solves it from a^2 + b^2 = c^2.
2. 2 Choose how to bridge to triangle 2: Enter a scale factor k directly, or pick a bridge side of triangle 2 you already know and type its length in the bridge value field.
3. 3 Read the solved sides of triangle 2: The tool reports the matched leg a, leg b, hypotenuse c, and final scale factor, so you can confirm the second triangle satisfies the Pythagorean theorem.
4. 4 Read the matching acute angles: Both triangles share the same acute angles, reported in degrees for slope, pitch, or bearing calculations.
5. 5 Check the area and area ratio: The area of each triangle and the area ratio triangle 2 / triangle 1 are returned; both should equal k^2 for any pair.
6. 6 Reset to the 3-4-5 example: Press Reset to reload the 3-4-5 reference with k = 2, which gives a 6-8-10 second triangle as a quick sanity check.

Sample run: Triangle 1 has legs 9 and 12 (hypotenuse 15), and you measured a leg of 6 on the second triangle. With bridge side = leg a and bridge value = 6, the tool recovers k = 6/9 = 0.6667 and prints b2 = 8, c2 = 10, angle opposite leg a = 36.87°, area ratio = 0.4444.

When the scale factor is given as a model ratio such as 1:24 or 1:87, the Scale Conversion Calculator converts the ratio into the same k you would type here.

The tool removes the manual cross-multiplication similar-triangle problems usually require and surfaces the angles and area ratio for free.

• • Two solution paths in one place: Enter the scale factor or recover it from any shared side; the tool decides which is shorter.
• • Auto-fills the missing hypotenuse: Leave the hypotenuse of triangle 1 blank and the tool solves a^2 + b^2 = c^2.
• • Returns the matching acute angles: Both triangles share the same acute angles; the tool prints them in degrees for slope, pitch, or bearing work.
• • Surfaces the area ratio for free: Areas scale by k^2, so the area of each triangle and the area ratio are returned for cost, weight, and material estimates.
• • Validates triangle 1 before solving: Entries where the hypotenuse is not the longest side, the sides break the Pythagorean theorem, or the bridge is inconsistent with k are rejected.
• • Useful in school, field, and design work: Covers textbook homework, survey shadow problems, scaled model-to-real-world conversions, and rafter/run calculations.

Treat the area ratio as a self-check: it should always equal k^2, and any other number tells you the scale factor or bridge side is off.

When the picture is two angles and a side rather than two sides and a scale factor, the AAA Triangle Calculator uses the Law of Sines to find the missing sides, area, and perimeter of a single triangle.

Four factors and two practical caveats control how much you can trust the output.

• • The tool assumes the two right triangles are already known to be similar. Confirm the AA condition (matching right angle plus one matching acute angle) before relying on the result.
• • Area ratios assume the figures are in the same plane and orientation; reflected or rotated triangles keep the same side ratios but the area is unaffected, while position-based downstream work is.

The single biggest source of wrong answers in similar-triangle work is treating the two triangles as similar when only one acute angle matches. The AA similarity test is the cheapest insurance against that mistake.

According to Wikipedia (Similarity (geometry)), two triangles are similar when their corresponding angles are equal and the ratios of their corresponding sides are equal, which makes every pair of right triangles with one matching acute angle similar.

To confirm the side-to-side ratio between triangle 1 and triangle 2 with mixed lengths, the Ratio Calculator simplifies a:b pairs to lowest terms so you can spot when the scale factor is not what you expected.