Similar Triangles Calculator

Solve for any missing side in a pair of similar triangles.

Select the side you want to find and enter the lengths of the other three corresponding sides.

Practical Examples

Load an example to see how the calculator works.

Find Side A₂

findSideA2

Given A₁=3, B₁=4, and B₂=8, find the length of side A₂.

Side A₁ (Triangle 1): 3

Side B₁ (Triangle 1): 4

Side B₂ (Triangle 2): 8

Find Side B₂

findSideB2

Given A₁=5, B₁=12, and A₂=10, find the length of side B₂.

Side A₁ (Triangle 1): 5

Side B₁ (Triangle 1): 12

Side A₂ (Triangle 2): 10

Find Side A₁

findSideA1

Given B₁=7, A₂=9, and B₂=21, find the length of side A₁.

Side B₁ (Triangle 1): 7

Side A₂ (Triangle 2): 9

Side B₂ (Triangle 2): 21

Find Side B₁

findSideB1

Given A₁=2.5, A₂=10, and B₂=15, find the length of side B₁.

Side A₁ (Triangle 1): 2.5

Side A₂ (Triangle 2): 10

Side B₂ (Triangle 2): 15

Other Titles
Understanding Similar Triangles: A Comprehensive Guide
An in-depth look at the principles of similar triangles, their properties, and how to solve problems involving them.

What Are Similar Triangles?

  • Definition and Core Concepts
  • Criteria for Similarity (AA, SAS, SSS)
  • The Importance of Corresponding Sides
In geometry, two triangles are considered 'similar' if they have the same shape, but not necessarily the same size. This means that all their corresponding angles are equal, and their corresponding sides are in proportion. The ratio of the lengths of corresponding sides is called the 'scale factor'.
Criteria for Proving Similarity
There are three main ways to determine if two triangles are similar:
1. Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
2. Side-Angle-Side (SAS) Similarity: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, the triangles are similar.
3. Side-Side-Side (SSS) Similarity: If the corresponding side lengths of two triangles are proportional, the triangles are similar.

Real-World Analogies

  • A model car and the actual car are based on similar triangles in their design.
  • Architectural blueprints use scaling, which relies on the principles of similarity.
  • Using shadows to calculate the height of a tall object.

Step-by-Step Guide to Using the Similar Triangles Calculator

  • Selecting the Unknown Side
  • Inputting Known Values
  • Interpreting the Results
Our calculator simplifies the process of finding a missing side in a pair of similar triangles. The fundamental formula is based on the proportion: A₁/A₂ = B₁/B₂.
How It Works:
1. Choose What to Find: Use the dropdown menu to select which of the four sides (A₁, B₁, A₂, or B₂) you wish to calculate.
2. Enter the Knowns: Fill in the input fields for the three known side lengths. Ensure that you are entering the lengths for corresponding sides correctly.
3. Calculate and Analyze: Click the 'Calculate' button. The calculator will solve for the unknown side using the proportionality rule. The result will be displayed clearly.

Calculation Formulas

  • To find A₂, the formula is A₂ = (A₁ * B₂) / B₁
  • To find B₁, the formula is B₁ = (B₂ * A₁) / A₂

Real-World Applications of Similar Triangles

  • Architecture and Engineering
  • Art and Design
  • Navigation and Surveying
The concept of similar triangles is not just an academic exercise; it has numerous practical applications in various fields.
Practical Examples:
Surveying: Surveyors use similar triangles to measure distances that are difficult to access directly, such as the width of a river or the height of a mountain.
Art and Perspective: Artists use similarity to create the illusion of depth and distance on a flat surface. Objects that are further away are drawn smaller, creating a set of similar figures.
Optics: Lenses and mirrors form images using the principles of similar triangles to determine the size and location of the image.

Application Scenarios

  • Calculating the height of a flagpole by measuring its shadow and the shadow of a person of known height.
  • Scaling a recipe up or down involves proportional relationships, similar to triangles.
  • Creating maps and scale models.

Common Misconceptions and Correct Methods

  • Confusing Congruence with Similarity
  • Incorrectly Matching Corresponding Sides
  • Mathematical Errors in Proportions
There are several common pitfalls to avoid when working with similar triangles.
Key Mistakes to Avoid:
Similarity vs. Congruence: Congruent triangles are identical in both shape and size (a scale factor of 1). Similar triangles only need to have the same shape. All congruent triangles are similar, but not all similar triangles are congruent.
Matching Sides: The biggest source of error is often mismatching corresponding sides. Side A₁ must correspond to A₂, and B₁ must correspond to B₂. Mixing them up (e.g., A₁/B₂ = B₁/A₂) will lead to incorrect results. Always identify which sides are opposite to which equal angles.

Correction Tips

  • Incorrect: A₁/B₂ = A₂/B₁. Correct: A₁/A₂ = B₁/B₂.
  • Always double-check which side is opposite which angle to ensure they correspond.

Mathematical Derivation and Formulas

  • The Proportionality Theorem
  • Deriving the Cross-Multiplication Formula
  • Understanding Scale Factor
The calculations performed by this tool are based on the fundamental property of similar triangles: the ratio of corresponding sides is constant.
The Core Formula
Let Triangle 1 have sides A₁ and B₁, and Triangle 2 have corresponding sides A₂ and B₂. If the triangles are similar, then:
A₁ / A₂ = B₁ / B₂
This is the 'proportionality equation'. To solve for any unknown side, we can rearrange this equation using simple algebra, a technique often called 'cross-multiplication'.
Derivations:
  • To find A₂, we rearrange to get: A₂ = A₁ * (B₂ / B₁)
  • To find B₂, we rearrange to get: B₂ = B₁ * (A₂ / A₁)
  • To find A₁, we rearrange to get: A₁ = A₂ * (B₁ / B₂)
  • To find B₁, we rearrange to get: B₁ = B₂ * (A₁ / A₂)

Formula Examples

  • If A₁=2, A₂=4, B₁=3, then B₂ must be 6 because 2/4 = 3/6.
  • The ratio A₁/A₂ is the scale factor from Triangle 2 to Triangle 1. In the example above, it's 2/4 = 0.5.