Euclid, through propositions, introduces two ideas that are nearly alike, equality and similarity. Equality and similarity are both applied across many concepts in Euclid's Elements. They both relate rectilineal figures¹ to other rectilineal figures, and circles to other circles.

All circles are similar to one another. This is because the definition of a circle allows that only one quality of a circle change between any circles. Only the radius varies from circle to circle. Equality among circles is very limited for the same reason. If the radii are equal, the circles are equal.

The nature of rectilineal figures begets more rules for achieving equality or similarity. Rectilineal figures are less specifically defined, therefore to achieve similarity, the sides must be of the same number between figures, corresponding sides must be proportional, and the angles contained by corresponding sides must be equal. The figures do not have to be equal to be similar.

In rectilineal figures, equality has a broader definition than in circles. If the figures are equal, the figures are equal. What does this mean? Euclid defines figure as that which is contained by any boundary or boundaries, but many, while studying Euclid lose sight of this. Most commonly this is called the area of a figure, or megathos. So if the areas are equal between two figures, the figures are equal².

How then are figures equal? Similarity combines with equality in rectilineal figures in a way that is not obvious for circles, and this leads to a better understanding of equality. Similar and equal rectilineal figures will coincide with on another. If one imagines two rectilineal figures that fit all rules of similarity and that are equal, matching any of the sides coinciding with its corresponding side in the other figure results in only one visible figure.

Now if one imagines two equal figures that are not similar, the concept of equality can be tangentially grasped. If one keeps the area or size of the two figures constant, but makes them similar, they will coincide. This is what it means to have equal figures. In circles this problem with understanding equality never arises. Any circle that is equal to another circle will always coincide with that circle, because all circles are similar.