Regular and irregular shapes

Some shapes are perfectly even: every side is the same length and every corner is the same. Those are regular shapes, and everything else is irregular. A shape can look neat and still be irregular, so the only way to be sure is to check two things, the sides and the angles.

The rule

A shape is regular when all its sides are the same length and all its angles are the same size. If either one is not true, the shape is irregular.

How to tell if a shape is regular

Check the sides. Are they all the same length? On a drawing, matching dashes across the sides mean matching lengths. If there are no dashes, measure the sides with a ruler.
If any side is longer or shorter than the others, you can stop. The shape is irregular.
Now check the angles, the corners. Are they all the same size, all as pointed or as open as each other? On a drawing, matching arcs mean matching angles, and a small square means a right angle.
If any angle is different, the shape is irregular. If the sides all match and the angles all match, it is regular.

Worked examples

Is this hexagon regular?

The one dash on every side means all six sides are the same length, and the one arc in every corner means all six angles are the same. Both checks pass, so it is a regular hexagon, the shape of a honeycomb cell.

Is a rectangle regular?

The angles pass: all four are right angles. The sides do not. The two long sides (one dash) are longer than the two short ones (two dashes). One check fails, so a rectangle is irregular.

Is this rhombus regular?

The sides pass: all four have the same dash, so they are the same length. The angles do not. Two corners are narrow (one arc) and two are wide (two arcs). So a rhombus is irregular, even though all its sides are equal.

Which of these pentagons is regular?

A. Both have 5 sides, so both are pentagons. A has five equal sides and five equal angles. B has one short side at the top and corners of different sizes, so B is an irregular pentagon.

Why you need both checks

Equal sides do not make the angles equal, and equal angles do not make the sides equal. Push a square over and you get a rhombus, with the same four sides but two narrow corners and two wide ones. Stretch a square and you get a rectangle, with the same four right angles but two long sides and two short ones. Each passes one check and fails the other.

The square is the four-sided shape that passes both, which makes it the only regular quadrilateral. A rectangle or a rhombus that is not a square is always irregular.

Triangles are the one shape where a single check is enough. A triangle with three equal sides always has three equal angles, 60 degrees each, and the other way round. It is called an equilateral triangle, and it is the only regular triangle.

Where mistakes happen

Checking only the sides. A rhombus has four equal sides, so it is easy to call it regular, but its angles are not all the same.
Checking only the angles. A rectangle has four right angles, but two of its sides are longer than the other two.
Calling a shape irregular because it is turned or tipped over. A square standing on one corner is still a square, and still regular. Turning a shape does not change its sides or its angles.
Thinking irregular means messy or wrong. A rectangle is perfectly neat and still irregular. Irregular only means the sides or the angles are not all the same.

Questions people ask

Is a square regular?

Yes. All four sides are the same length and all four angles are right angles. It is the only regular shape with four sides.

Is a circle regular?

Regular and irregular are words for shapes with straight sides, called polygons. A circle has no straight sides and no corners, so the question does not really apply to it.

Does the size of a shape matter?

No. A tiny regular hexagon and a huge one are both regular. Regular is about the sides and the angles matching each other, not about how big the shape is.

Where can you see regular shapes?

A stop sign is a regular octagon, and the cells of a honeycomb are regular hexagons. The squares on a chessboard are regular too, but the rectangles of a brick wall are not.