Solve Similar Triangles Calculator - Proportional Sides Solver

A solve similar triangles calculator finds the missing side of a similar triangle using the proportional-sides property: when two triangles share the same three angles, the ratios of their matching sides are all equal. Enter two corresponding side pairs to set the scale factor, then enter the side whose match you need, and the calculator returns the missing side, the scale factor, and a similarity check that flags non-corresponding pairs. It is the natural tool for homework, blueprint scaling, map distances, and any geometry problem where you know some side lengths on two triangles of the same shape.

• • Geometry homework: Solve textbook problems where you are given two corresponding sides and asked for a third.
• • Map and blueprint scaling: Convert measurements between a drawing and the real object when the scale ratio is known from a labeled distance.
• • Model and photograph work: Translate measured lengths from a photograph or scale model back to the real-world size using one known reference length.
• • Construction drawing check: Verify that two parts of a drawing are actually the same shape by comparing two independent side ratios.

The only thing the user has to be careful about is identifying which side on the second triangle corresponds to which side on the first, which is why the calculator shows the second ratio as a similarity check. Once the two ratios agree, every unknown side on the second triangle is the corresponding side on the first multiplied by the scale factor, and the form does that multiplication in real time.

For the general side-side-side, side-angle-side, and angle-side-angle workflows that complement the similar-triangles case, the triangle calculator handles all of the other common triangle-solving paths in one place.

The calculator reads the two corresponding side pairs, computes the scale factor k = sideA2 / sideA1, computes a second ratio from the b pair, and then multiplies the unknown side c1 by k to recover side c2. The second ratio acts as a built-in similarity check.

• a1, b1, c1: Side lengths on the first triangle. Each must match a side on the second triangle that sits opposite the equal angle.
• a2, b2, c2: Corresponding side lengths on the second triangle. c2 is the unknown the calculator returns.
• k: The single scale factor that links the two triangles, equal to any one of the three ratios.

The second ratio is what catches the most common mistake: pairing up the wrong sides. If the user enters sides that are not actually corresponding, k and the b ratio will disagree, the similarity check returns mismatch, and the user knows to revisit which side on the second triangle matches which side on the first.

According to Wikipedia, two triangles are similar if and only if their corresponding angles are equal and the ratios of the lengths of corresponding sides are all equal to the same scale factor.

When the scale factor you computed is meant to translate between two systems of measurement (for example a 1:25,000 map or a 1:87 model scale), the scale conversion calculator lets you switch the result into a different unit without re-doing the similar-triangle math.

These four ideas cover what makes a pair of triangles similar, why a single scale factor is enough to recover every missing side, and how the calculator uses them in one pass.

These four ideas are the smallest toolkit you need to attack any similar-triangle problem. The AA test tells you when the rule applies, the scale factor collapses the size information into one number, and the proportional-sides rule does the actual side arithmetic.

The proportional-sides rule here is the same a/b = c/d logic the ratio calculator uses, so if you only need to simplify or check a single ratio without the triangle labels, it is the cleaner tool.

Work in order: first set up the reference pair that defines the scale, then confirm the similarity, then ask for the missing side. The result panel updates as you type.

1. 1 Enter side a on triangle 1: Type the length of the first reference side on the first triangle. Use the same length unit for every input.
2. 2 Enter side a on triangle 2: Type the length of the side on the second triangle that corresponds to side a. This fixes the scale factor k.
3. 3 Enter side b on triangle 1: Type the length of a second known side on triangle 1 whose match on triangle 2 you can also measure.
4. 4 Enter side b on triangle 2: Type the side on triangle 2 that corresponds to side b. The calculator divides b2 by b1 to produce the second ratio.
5. 5 Enter side c on triangle 1: Type the side on triangle 1 whose matching side on triangle 2 is unknown. The result panel shows side c on triangle 2 as k × c1.
6. 6 Read the scale factor and check: Use the scale factor k to plan further work and confirm the similarity check reads match before trusting the missing side.

For a 1:25,000 map, enter sideA1 = 1.4, sideA2 = 35,000, sideB1 = 1, sideB2 = 25,000, then sideC1 equal to the next map distance. The calculator returns k = 25,000 and c2 = 25,000 × c1, so you can keep measuring in one workflow.

When the measurements you actually have are angles and a single side rather than two side pairs, the AAA triangle calculator takes the same a reference side and derives the other two sides from the angle inputs.

Working the proportion by hand is fast once you know the rule, but the calculator removes a few error classes that show up over and over in real problems.

• • Built-in similarity check: Because the calculator shows the second ratio next to the scale factor, you can see at a glance whether the two triangles really match.
• • One workflow for any case: The same form works when the second triangle is larger, smaller, or congruent; the scale factor is just a number greater than, less than, or equal to 1.
• • No unit conversion step: Side lengths can be in centimeters, inches, feet, or any other length unit as long as all five inputs use the same one.
• • Real-time updates: The result panel recalculates as you type, so it is easy to try several guesses until the ratios agree.
• • Catches non-corresponding sides: When the b ratio disagrees with the a ratio, the calculator labels the result as a mismatch so you can fix the pairing first.

These benefits matter most when the inputs come from a real measurement rather than a textbook problem. On a construction drawing or a map, the labeling is what determines which sides correspond, and the similarity check is what catches a wrong label.

When your data is two angles and a non-included side rather than two corresponding side pairs, the AAS triangle calculator completes the triangle with the Law of Sines, which is the angle-first counterpart to this side-first similar-triangles workflow.

Four things move the answer the calculator returns, and the limitations below cover the assumptions behind the proportional-sides rule itself.

• • The proportional-sides rule is for planar Euclidean triangles. It does not apply to spherical triangles, where the side lengths no longer scale by a single constant and a different set of identities is required.
• • Two pairs of equal angles are needed to confirm that the triangles are similar. The calculator can flag a mismatch, but it cannot prove the triangles are similar; only the angle measurements or an explicit AA/SSS/SAS similarity check can.

These factors and limitations explain why the form takes a second ratio even when the user already has the missing side in mind: the extra pair makes the calculator trustworthy.

According to Wolfram MathWorld, if two triangles are similar then the ratios of the lengths of their corresponding sides are all equal, so a single scale factor relates every pair of matching sides.

After the calculator confirms the two triangles are similar, you can use the triangle area calculator to find the area of either triangle from its three sides, which gives a quick cross-check on the scale factor (area scales as the square of k).