Solve Similar Triangles
Solve similar triangles by finding the missing side length using the scale factor between proportional sides. Free geometry tool for students and teachers.
About This Calculator
The Solve Similar Triangles Calculator helps you find the missing side length in a pair of similar triangles. Similar triangles are triangles that have the same shape but different sizes -- their corresponding angles are equal and their corresponding sides are proportional. This free online geometry tool is ideal for students learning triangle similarity, teachers preparing geometry lessons, and professionals working with scaled drawings and proportional measurements.
How the calculation works
The calculator uses the fundamental property of similar triangles: corresponding sides are in the same ratio. When you enter two side lengths from Triangle 1 and the matching side from Triangle 2, the calculator computes the scale factor by dividing the Triangle 2 side by its corresponding Triangle 1 side. It then multiplies the other known side in Triangle 1 by this scale factor to find the missing side length in Triangle 2.
Mathematically, if side A in Triangle 1 measures a1 and its corresponding side A in Triangle 2 measures a2, the scale factor k = a2 / a1. If side B in Triangle 1 measures b1, then the missing side B in Triangle 2 is b2 = b1 x k. This formula works for any pair of corresponding sides between similar triangles.
Common similarity criteria
Two triangles are similar if they satisfy the SSS criterion (all three sides proportional), SAS criterion (two sides proportional with equal included angle), or AA criterion (two angles equal). The AA criterion is the most common because if two angles are equal, the third is automatically equal since all triangles have 180 degrees. This calculator assumes you already know the triangles are similar and focuses on finding the missing measurement.
Applications of similar triangles
Similar triangles appear throughout geometry, trigonometry, and real-world applications. Architects use similar triangles for scale drawings and model building. Engineers apply similarity principles in map scaling, optical systems, and structural analysis. Surveyors use similar triangles to measure distances indirectly. Photography, computer graphics, and display scaling all rely on the proportional relationships that define similar triangles.
Frequently Asked Questions
How does the Solve Similar Triangles calculator work?
The calculator finds the missing side in a pair of similar triangles by computing the scale factor. Enter two known sides from Triangle 1 and the matching side from Triangle 2. The calculator divides the matching side in Triangle 2 by its corresponding side in Triangle 1 to get the scale factor, then multiplies the other known side in Triangle 1 by this scale factor to find the missing side.
What are similar triangles?
Similar triangles are triangles that have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. This means one triangle is a scaled version of the other. The ratio between any two corresponding sides is called the scale factor.
How do you find the scale factor of similar triangles?
To find the scale factor, divide the length of any side in Triangle 2 by the length of its corresponding side in Triangle 1. For example, if side A in Triangle 1 is 6 units and side A in Triangle 2 is 9 units, the scale factor is 9 divided by 6, which equals 1.5. You can then multiply this scale factor by any side in Triangle 1 to find the corresponding side in Triangle 2.
What criteria prove two triangles are similar?
Two triangles are similar if they satisfy any one of three criteria: SSS (all three corresponding sides are in the same proportion), SAS (two corresponding sides are in proportion and their included angles are equal), or ASA (two corresponding angles are equal and the sides between them are in proportion). The Angle-Angle (AA) criterion is sufficient since two equal angles automatically make the third equal.
Can this calculator handle right triangles?
Yes, this calculator works for right triangles as well as any other type of triangle. The same similarity rules apply. For right triangles specifically, you can compare the legs and hypotenuse between two similar right triangles using the same scale factor method.
Is this similar triangles tool free to use?
Yes, all calculators on Calculy are completely free to use. There are no hidden charges, subscriptions, or usage limits. You can use this similar triangles solver as many times as you need for your geometry homework, teaching, or professional work.
What units should I enter for the triangle sides?
You can enter side lengths in any unit as long as you are consistent across all three inputs. The calculator works with any unit of length -- centimeters, meters, inches, feet, or any other measurement. The scale factor is a dimensionless ratio, so the missing side will be in the same unit as your input values.