Similar Triangles Calculator

Category: Geometry
- April 13, 2025
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Calculate and verify similarity between triangles using different criteria such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Angle (AA), and find unknown side lengths and angles.

Similarity Criteria Selection

Use the properties of similar triangles to find an unknown side length.

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Display Options

About Similar Triangles

Similar triangles have the same shape but not necessarily the same size. Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

Similarity Criteria
  • Angle-Angle (AA): Two triangles are similar if two angles of one triangle are equal to two angles of another triangle. (The third angle will also be equal.)
  • Side-Side-Side (SSS): Two triangles are similar if all their corresponding sides are proportional.
  • Side-Angle-Side (SAS): Two triangles are similar if two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal.
Properties of Similar Triangles
  • Corresponding angles are equal.
  • Corresponding sides are proportional.
  • The ratio of corresponding sides is called the scale factor (or similarity ratio).
  • The ratio of the areas of similar triangles is equal to the square of the scale factor.
  • The ratio of the perimeters of similar triangles is equal to the scale factor.
Applications of Similar Triangles

Similar triangles have numerous practical applications, including:

  • Finding heights of tall objects indirectly
  • Solving real-world proportionality problems
  • Map scaling and distance calculations
  • Architecture and engineering design
  • Optical systems and lens calculations

Key Formula for Similar Triangles:

Corresponding sides are proportional:

\( \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \)

Corresponding angles are equal:

\( \angle A = \angle D, \quad \angle B = \angle E, \quad \angle C = \angle F \)

What Is the Similar Triangles Calculator?

The Similar Triangles Calculator is a simple yet effective tool that helps you solve triangle measurements based on triangle similarity rules. Whether you're verifying if two triangles are similar or calculating missing side lengths or angles, this triangle geometry helper simplifies the process.

How This Tool Can Help

This calculator is especially useful for students, teachers, and anyone needing to:

  • Determine if two triangles are similar using criteria like Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Angle (AA).
  • Find unknown side lengths when triangle similarity is known.
  • Calculate unknown angles based on matching triangle structures.
  • Identify the scale factor between two similar triangles.

It acts as a right triangle solver in specific cases, and complements other tools like the triangle side and angle tool, area calculator, or even a Pythagorean theorem tool.

How to Use the Calculator

Start by choosing your calculation type from the dropdown menu:

  • Find Unknown Side Length – Input corresponding side values from each triangle to solve the missing one.
  • Verify Triangle Similarity – Use the SSS, SAS, or AA method to check if two triangles are similar.
  • Find Scale Factor – Input all side lengths from both triangles to determine how they scale.
  • Find Unknown Angle – Use known angles from one triangle to determine missing values in a similar one.

After entering the values, hit "Calculate" to get the results. Use the optional settings to adjust decimal places or view detailed steps.

Why Similar Triangles Matter

Similar triangles are widely used across geometry problems. Knowing how to solve them allows you to:

  • Estimate heights and distances indirectly
  • Simplify geometric proofs
  • Understand how shapes relate through proportion
  • Support calculations for architecture, design, and engineering tasks

This triangle solver works well in combination with tools like a right triangle calculator, distance calculator, or volume calculator depending on the context of the problem.

Frequently Asked Questions (FAQ)

What is triangle similarity?

Triangle similarity means two triangles have the same shape, with corresponding angles equal and corresponding sides in the same ratio.

What are the criteria to confirm triangle similarity?

You can confirm similarity using:

  • AA (Angle-Angle): Two matching angles
  • SSS (Side-Side-Side): All corresponding sides proportional
  • SAS (Side-Angle-Side): Two sides in proportion and included angles equal

Can I use this calculator for right triangles?

Yes. While this is not limited to right triangles, it complements a right triangle solver or hypotenuse finder when dealing with specific right-angle cases.

What is the scale factor?

The scale factor is the ratio of a side in one triangle to its corresponding side in a similar triangle. It also determines the ratio of their perimeters and the square root of the ratio of their areas.

Can this calculator help with triangle construction?

Yes, it's a reliable triangle geometry helper that can support constructing or comparing triangles in both academic and practical settings.

Explore More Geometry Tools

This Similar Triangles Calculator is part of a broader set of geometry tools. For other shapes and measurements, you might find these helpful:

These tools help you solve geometry problems faster and with more confidence.