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Triangle Similarity Calculator

Check if two triangles are similar by comparing their side ratios (SSS similarity test). Enter three sides of each triangle to determine if corresponding sides are proportional and find the scale factor, essential for geometry proofs and real-world scaling problems.

Reviewed by Christopher FloiedPublished Updated

This free online triangle similarity calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Minimum: 0.01

First side of triangle 1

Minimum: 0.01

Second side of triangle 1

Minimum: 0.01

Third side of triangle 1

Minimum: 0.01

First side of triangle 2

Minimum: 0.01

Second side of triangle 2

Minimum: 0.01

Third side of triangle 2

Results

Scale Factor (T2/T1)

2

Similar (1=Yes, 0=No)

1

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Triangle Similarity Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Triangle Similarity Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Triangle Similarity Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Triangle Similarity Calculator

The Triangle Similarity Calculator checks whether two triangles are similar using the SSS (Side-Side-Side) similarity criterion and computes the scale factor. Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. The SSS similarity test checks that the ratios of sorted sides are all equal. Similar triangles are fundamental in geometry: they underlie proportional reasoning, the definition of trigonometric functions, indirect measurement techniques, and scaling in engineering and architecture. Map-making, shadow measurements, and photographic enlargement all rely on triangle similarity.

The Math Behind It

Two triangles are similar if they have the same shape but possibly different sizes. This means all corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor). There are three similarity criteria: SSS Similarity: If the ratios of all three pairs of corresponding sides are equal (a1/a2 = b1/b2 = c1/c2 after matching smallest to smallest, etc.), the triangles are similar. AA Similarity: If two pairs of corresponding angles are equal, the triangles are similar. (The third pair automatically matches since angles sum to 180.) SAS Similarity: If two pairs of sides are proportional and the included angles are equal, the triangles are similar. The scale factor k = (side in triangle 2)/(corresponding side in triangle 1) relates all linear dimensions. Areas scale by k^2, and if extended to 3D, volumes scale by k^3. This scaling relationship is fundamental in physics: it explains why ants can carry many times their body weight (strength scales as k^2 but weight as k^3), why elephants cannot jump, and why miniature models behave differently from full-scale structures. Thales of Miletus (circa 600 BCE) used triangle similarity to measure the height of the Great Pyramid from its shadow. Eratosthenes used similar triangles to estimate Earth's circumference. In modern applications, similar triangles are used in photogrammetry, optical systems, and computer vision for depth estimation from stereo images.

Formula Reference

SSS Similarity Test

a1/a2 = b1/b2 = c1/c2 (after sorting corresponding sides)

Variables: Sides of triangle 1 and triangle 2 sorted in ascending order

Worked Examples

Example 1: Similar Triangles: 3-4-5 and 6-8-10

Check if triangles (3,4,5) and (6,8,10) are similar.

Step 1:Sort T1: 3, 4, 5. Sort T2: 6, 8, 10
Step 2:Ratios: 6/3 = 2, 8/4 = 2, 10/5 = 2
Step 3:All ratios equal: triangles are similar with scale factor 2

Similar with scale factor 2.

Example 2: Non-Similar Triangles: 3-4-5 and 3-5-7

Check if triangles (3,4,5) and (3,5,7) are similar.

Step 1:Sort T1: 3, 4, 5. Sort T2: 3, 5, 7
Step 2:Ratios: 3/3 = 1, 5/4 = 1.25, 7/5 = 1.4
Step 3:Ratios are not equal: triangles are NOT similar

Not similar (ratios differ: 1, 1.25, 1.4).

Common Mistakes & Tips

  • !Comparing sides in the wrong order. Always sort both triangles' sides (smallest to largest) before comparing ratios.
  • !Confusing similarity with congruence. Similar triangles have the same shape but can be different sizes. Congruent triangles are both the same shape and size (scale factor = 1).
  • !Expecting exact floating-point equality. Due to rounding, ratios may differ by tiny amounts. Use a small tolerance (like 0.001) when comparing.

Related Concepts

Triangle Congruence

Congruence is similarity with scale factor 1. Congruent triangles have equal sides AND equal angles.

Scalene Triangle Area

Similar triangles have areas proportional to the square of the scale factor. If k = 2, the larger triangle has 4 times the area.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Triangle Congruence Calculator

Frequently Asked Questions

How does the area of similar triangles relate?

If two triangles are similar with scale factor k, the ratio of their areas is k^2. For example, if one triangle has sides twice as long as another (k=2), its area is 4 times larger.

Can I determine similarity from just two sides?

Not with sides alone. You need either all three side ratios (SSS), two sides and the included angle (SAS), or two angles (AA). Two sides without the angle between them are insufficient.

How did ancient Greeks use triangle similarity?

Thales measured the height of the Great Pyramid by comparing his shadow length to the pyramid's shadow. Since the sun's rays create similar triangles, the ratio of shadow to height is the same for both. Eratosthenes used similar geometry to estimate Earth's circumference by comparing shadow angles at two cities.

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