Similar Triangles Calculator
Enter the sides of two similar triangles to find the missing side. Similar triangles have proportional sides, and this tool uses that ratio to solve for unknowns.
Understanding Similar Triangles
Two triangles are similar if they have the same shape, meaning all corresponding angles are equal. When angles match, the sides automatically become proportional. The ratio of any two corresponding sides is constant across the triangles.
For example, if one triangle has sides 3, 4, 5 and another has sides 6, 8, 10, the scale factor is 2. Each side of the larger triangle is exactly twice the corresponding side of the smaller triangle. This proportionality extends to all three pairs of sides.
This calculator exploits that proportionality. Enter two sides of the first triangle and the corresponding side of the second triangle, and it computes the missing side by multiplying by the scale factor. It also displays the scale factor explicitly so you can verify the relationship.
Practical Uses for Similar Triangles
Surveyors and architects use similar triangles to measure heights indirectly. By measuring shadows or using a small reference triangle, they can calculate the height of buildings, trees, or mountains without climbing them. The principle is simple: set up proportions between the known and unknown sides.
Scale models rely on similar triangles. A model airplane might use a 1:100 scale, meaning every triangle in the model is similar to the corresponding triangle in the real plane with a scale factor of 1/100. Designers verify dimensions by checking proportionality.
In optics, similar triangles describe how lenses and mirrors form images. The object distance, image distance, and focal length form proportional relationships that determine image size and position.
Similar Triangles vs. Congruent Triangles
Congruent triangles are identical in both shape and size. All corresponding sides and angles are equal. Similar triangles match in shape but can differ in size. Congruence is a special case of similarity where the scale factor equals 1.
For congruent triangles, side ratios are all 1:1. For similar triangles, the ratios can be any positive number. This flexibility makes similarity more common in real-world applications, where exact duplication is rare but proportional scaling is frequent.
This calculator handles the general case of similar triangles. If your triangles are congruent, enter them as similar triangles with a scale factor of 1, and the missing sides will match the known sides exactly.
Frequently Asked Questions
What are similar triangles?
Similar triangles have the same shape but different sizes. All corresponding angles are equal, and all corresponding sides are proportional.
How do you find a missing side in similar triangles?
Set up a proportion using corresponding sides. If triangle 1 has sides a and b, and triangle 2 has side a' known, then b' = b × (a' / a).
What is the scale factor?
The scale factor is the ratio of corresponding sides in the two triangles. It tells you how many times larger (or smaller) one triangle is compared to the other.
Do similar triangles have the same area?
No. The area ratio equals the square of the scale factor. If the scale factor is 2, the larger triangle has 4 times the area of the smaller one.
How do you prove two triangles are similar?
You can prove similarity by showing all corresponding angles are equal (AA criterion), or all corresponding sides are proportional (SSS criterion), or two sides are proportional and the included angle is equal (SAS criterion).