Similar Triangles Solver Calculator

What Are Similar Triangles?

Two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional. This proportionality is expressed through a single multiplier called the scale factor (k), which relates every side of one triangle to the corresponding side of the other. Similarity is one of the most powerful concepts in Euclidean geometry, enabling mathematicians, engineers, and surveyors to calculate unknown distances without direct measurement.

The Core Formula

For two similar triangles with sides a, b, c (Triangle 1) and a′, b′, c′ (Triangle 2), the foundational proportion is:

a′/a = b′/b = c′/c = k

The scale factor k equals a′ divided by a. Once k is established, any unknown corresponding side is found by multiplying the known Triangle 1 side by k. To solve for a missing side b′:

b′ = b × (a′ / a)

Deriving the Scale Factor: Step by Step

First, identify one pair of corresponding sides — one value from each triangle. Divide the Triangle 2 side by its Triangle 1 counterpart to compute k. Then multiply any other Triangle 1 side by k to obtain the Triangle 2 equivalent.

Worked Numeric Example

Triangle 1 has sides a = 5, b = 12, and c = 13 (a classic Pythagorean triple). Triangle 2 is similar with a′ = 10. Computing the scale factor: k = 10 ÷ 5 = 2. Therefore b′ = 12 × 2 = 24 and c′ = 13 × 2 = 26. Triangle 2 has sides 10, 24, and 26 — exactly twice the dimensions of Triangle 1, preserving all angles.

Similarity Criteria

Three standard criteria confirm triangle similarity without requiring every measurement. AA (Angle-Angle): two pairs of corresponding angles are equal. SAS (Side-Angle-Side): two proportional sides enclose an equal included angle. SSS (Side-Side-Side): all three pairs of corresponding sides are proportional. According to Maricopa Open College Mathematics, the AA postulate is the most widely applied criterion in both academic and practical contexts, because confirming two angles is typically far simpler than measuring and comparing all three side pairs.

Real-World Applications

• Surveying: Surveyors determine inaccessible distances — such as a river's width — by constructing a similar triangle on accessible ground. If the known base measures 30 m and k = 1.5, the inaccessible span is 30 × 1.5 = 45 m, with no need to cross the obstacle.
• Architecture and Scale Drawings: A blueprint at 1:100 scale applies k = 0.01. Every paper measurement multiplied by 100 yields the real-world dimension, allowing precise construction from compact drawings.
• Shadow Problems: A 6-foot person casting a 4-foot shadow creates a similar triangle with the ground and the light source. A nearby pole casting a 14-foot shadow stands 6 × (14 ÷ 4) = 21 feet tall — no ladder required.
• Related Rates in Calculus: As detailed in Baruch College's calculus tutorial on shadows and similar triangles, the same proportionality extends to dynamic scenarios, such as computing how fast a shadow lengthens as a person walks away from a lamppost at a known speed.
• Optics and Photography: The ratio of image height to object height equals the ratio of image distance to object distance — a direct application of similar triangle scaling used in lens design, projectors, and camera optics.

Calculator Variables Explained

• a — Triangle 1 Known Side: A positive reference side from the first triangle. Acts as the denominator of the scale ratio. Must be greater than 0.
• a′ — Triangle 2 Corresponding Side: The side in Triangle 2 that directly corresponds to side a. Dividing a′ by a yields the scale factor k.
• b — Triangle 1 Other Side: A second known side from Triangle 1. The calculator multiplies b by k to produce b′, the missing corresponding side in Triangle 2.
• k — Scale Factor: The ratio k = a′/a. Values greater than 1 indicate Triangle 2 is larger; less than 1 means smaller; exactly 1 means the triangles are congruent.

Assumptions and Limitations

This calculator assumes the two triangles are already confirmed to be similar — it does not derive similarity from angle measurements alone. All input values must be positive real numbers. The proportionality formula applies equally to any pair of similar polygons, not just triangles, provided corresponding sides are correctly identified and matched before entry.