Regular polygon facts for kids

This picture shows a regular polygon with 5 sides, called a pentagon. It has equal sides (s), a circumradius (R), and an apothem (a).

A square is a regular polygon. Its diagonals are shown here.

In geometry, a regular polygon is a special type of polygon. What makes it special? It has two main features:

• All its sides are the same length (this is called equilateral).
• All its angles are the same size (this is called equiangular).

Think of a square or a stop sign (which is an octagon). These are perfect examples of regular polygons!

Regular polygons can be either convex or star. A convex polygon bulges outwards, like a normal square or triangle. A star polygon has points that go inwards and outwards, like a five-pointed star.

If you imagine a regular polygon with more and more sides, it starts to look more and more like a circle. If you keep adding sides forever, it would eventually become a perfect circle!

What Makes Regular Polygons Special?

Here are examples of regular convex and star polygons. They have different numbers of sides, from 3 to 12.

Regular polygons have some cool properties that are always true, whether they are convex or star-shaped.

Symmetry and Circles

• Rotational Symmetry: A regular polygon with n sides has n-fold rotational symmetry. This means you can turn it around its center n times, and it will look exactly the same each time before it completes a full circle. For example, a square (4 sides) looks the same if you turn it 90 degrees.
• Circumscribed Circle: All the corners (called vertices) of a regular polygon always lie on a single circle. This circle is called the circumscribed circle. It's like you can draw a circle that perfectly touches every corner of the polygon.
• Inscribed Circle: Because all sides are the same length, every regular polygon also has a circle that fits perfectly inside it. This circle, called the inscribed circle, touches the middle of every side.

Building Regular Polygons

You can draw some regular polygons using only a compass and a straightedge (a ruler without markings). This is possible if the number of sides has special prime factors called Fermat primes. For example, you can draw a regular triangle (3 sides), a square (4 sides), or a pentagon (5 sides) this way. But you can't draw a regular heptagon (7 sides) with just a compass and straightedge.

Images for kids

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  This picture compares regular polygons with the same side length. They range from a triangle (3 sides) to a hexacontagon (60 sides). Notice how the polygon gets bigger as the number of sides increases!