Similar Triangles Calculator
Enter the sides (and angles where needed) of two triangles to check whether they are similar and to find any missing side length. The calculator supports three similarity criteria: Side-Side-Side (SSS), Side-Angle-Side (SAS), and Angle-Angle (AA). It returns the scale factor between the triangles, the area and perimeter ratios, and a complete step-by-step solution you can copy into your working.
Formula
Worked example
Triangle 1 has sides 3, 4, 5. Triangle 2 has sides 6, 8, 10. The ratios are 6/3 = 2, 8/4 = 2, 10/5 = 2, all equal to k = 2, so the triangles are similar by SSS. The area of Triangle 1 is 6 (right triangle), so the area of Triangle 2 is 6 * 2^2 = 24. The perimeters are 12 and 24, giving a perimeter ratio of 2, which equals k.
What are similar triangles?
Two triangles are similar when they have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are proportional. The ratio of any pair of corresponding sides is called the scale factor k. For example, a triangle with sides 3, 4, 5 and one with sides 6, 8, 10 are similar: every side in the second triangle is exactly twice the corresponding side in the first, giving k = 2. Similar triangles appear throughout geometry, trigonometry, and real-world applications such as using shadows to measure heights of trees or buildings.
The three main similarity criteria
You do not need to know every side and angle to prove similarity. Three criteria are sufficient: SSS (Side-Side-Side): All three pairs of corresponding sides are in the same ratio. If AB/DE = BC/EF = AC/DF = k, the triangles are similar. SAS (Side-Angle-Side): Two pairs of corresponding sides are proportional AND the angle between them (the included angle) is equal in both triangles. AA (Angle-Angle): Two pairs of corresponding angles are equal. Because angles in any triangle sum to 180 degrees, two equal angles automatically force the third to be equal too, so AA is often the easiest criterion to apply.
Scale factor, area ratio, and perimeter ratio
Once you know the scale factor k, you can derive every measurement of the second triangle from the first. Linear measurements (sides, perimeter, height, median) all scale by k: if Triangle 1 has perimeter P, Triangle 2 has perimeter k * P. Areas scale by k squared: if Triangle 1 has area A, Triangle 2 has area k^2 * A. Volumes of three-dimensional shapes scaled by k multiply by k cubed. These relationships are the reason similar triangles underpin scale drawings, maps, and architectural models.
Real-world uses of similar triangles
Similar triangles are one of the most practical tools in applied geometry. Surveyors use them to measure distances across rivers or ravines without crossing. Navigators have used them for centuries with instruments like the sextant. A person 1.8 m tall standing 5 m from a lamppost and casting a 3 m shadow on the ground can compute the height of the lamppost using similar triangles. Architects and engineers use scale factors to convert between drawings and real structures. Photographers use the same principle (a pinhole camera forms a similar triangle between the subject and the image plane).
Triangle similarity criteria
| Criterion | Abbreviation | What must match | Sides needed | Angles needed |
|---|---|---|---|---|
| Side-Side-Side | SSS | All three corresponding side ratios are equal | 3 pairs | 0 |
| Side-Angle-Side | SAS | Two corresponding side ratios equal AND the included angle is equal | 2 pairs | 1 shared |
| Angle-Angle | AA | Two pairs of corresponding angles are equal (third follows automatically) | 0 | 2 pairs |
| Angle-Side-Angle | ASA | Two angles equal and the side between them proportional (reduces to AA) | 1 pair | 2 pairs |
| Right-Angle-Hypotenuse-Side | RHS | Right angle, equal hypotenuse ratio, one other side ratio equal | 2 pairs | 1 (90 deg) |
The three standard criteria accepted by all geometry curricula. Only one needs to be satisfied.
Frequently asked questions
What is the difference between similar triangles and congruent triangles?
Congruent triangles are identical in both shape and size - every side and angle is exactly equal, so the scale factor k = 1. Similar triangles have the same shape (equal angles) but can be any size - their sides are proportional rather than equal. All congruent triangles are similar, but most similar triangles are not congruent.
Do I need all three side ratios to be equal, or just two?
For the SSS criterion you need all three ratios to be equal. However, if you have two equal angles (AA criterion), no side ratios are required at all. For SAS, two side ratios and one angle are enough. Two side ratios alone (without an angle) are not sufficient to guarantee similarity.
How do I find a missing side using similar triangles?
First establish the scale factor k from any pair of known corresponding sides (k = known side in T2 / corresponding known side in T1). Then multiply any side of Triangle 1 by k to get the corresponding side of Triangle 2. For example, if k = 2.5 and Triangle 1 has side c = 7, then Triangle 2 has side c = 2.5 * 7 = 17.5.
Why does the area ratio equal k squared?
Area is a two-dimensional measurement proportional to the square of any linear dimension. When all sides scale by k, the area scales by k * k = k squared. Intuitively, if you scale a shape up by a factor of 3 in every direction, you end up fitting 9 copies of the original inside the new shape (3 rows of 3).
Can right triangles be similar?
Yes. Any two right triangles that share one acute angle are similar by the AA criterion (the right angle is 90 degrees in both, and the shared acute angle is the second matching angle). The well-known 3-4-5 right triangle is similar to any triangle with sides in the ratio 3:4:5, such as 6-8-10 or 1.5-2-2.5.
How do I match corresponding sides correctly?
Corresponding sides are opposite to equal angles. Label the angles in both triangles and match the largest angle in Triangle 1 with the largest in Triangle 2, the smallest with the smallest, and the middle with the middle. The sides opposite each matched pair of angles are corresponding sides. A common mistake is matching sides by position in the diagram rather than by the angles they face.