Geometry Problem Solver What does similar mean in math?
Two figures are similar when they have the same shape but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are in the same ratio, called the scale factor. Congruent is the stricter case where the scale factor is exactly 1.
Similar or congruent
Congruent figures are identical: same shape and same size, so one can be placed exactly on the other after sliding, turning or flipping. Similar figures allow a change of size on top of that.
So every congruent pair is also similar, with a scale factor of 1, but similar pairs are usually not congruent. The symbols differ too: ~ means similar, ≅ means congruent.
Using the scale factor
The scale factor is the ratio of any pair of corresponding sides, and it is the same for all of them. Find it once from a pair you know, then use it on the side you want.
Watch out for area and volume, which do not scale by the same number. Double the sides and the area goes up by 2² = 4, and volume by 2³ = 8. A model at half scale has a quarter of the surface and an eighth of the volume.
Proving two triangles similar
Three shortcuts do it. AA: two pairs of equal angles are enough, because the third pair then matches automatically. SSS: all three pairs of sides in the same ratio. SAS: two pairs of sides in ratio with the angle between them equal.
AA is the one that does most of the work in practice, since parallel lines and shared angles hand you equal angles for free.
Similar against congruent
| Similar | Congruent | |
|---|---|---|
| Angles | Equal | Equal |
| Sides | In the same ratio | Equal |
| Size | May differ | Identical |
| Symbol | ~ | ≅ |
| Scale factor | Any positive number | Exactly 1 |
Questions
Are all squares similar?
Yes. Every square has four right angles and four equal sides, so any two squares differ only by a scale factor. The same holds for all circles and all equilateral triangles.
Are all rectangles similar?
No. Their angles always match, but the ratio of length to width can differ. A 2 by 1 rectangle and a 3 by 1 rectangle are not similar.
Does similar mean the same size?
No, that is congruent. Similar allows any change of size as long as the shape is preserved, which means equal angles and proportional sides.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
