Regular polygon

Some shapes are very even.

All their sides have the same length. All their corners look the same, too. These are called regular polygons.

Some look like stars.

These are regular star polygons. They are not flat or simple.

All the corners touch a circle.

You can draw a circle around them. This circle touches every corner.

You can also fit a circle inside. This small circle touches the middle of every side. These even shapes look very neat.

A regular polygon is a very even shape.

To be regular, all sides must have the same length. All the corners, or angles, must be the same size too. Some regular shapes are simple and flat. Others look like stars. These are called star polygons.

These shapes have special rules. All the corners touch a large circle. This is called a circumscribed circle. You can also fit a small circle inside. This circle touches the middle of every side. We call this an inscribed circle.

As you add more sides, the shape changes. A shape with many sides looks almost like a circle. For example, a shape with 10,000 sides is very round. Its corners are almost straight lines. However, a polygon can never truly become a circle.

Some shapes are easy to draw. You can make them using just a compass and a straightedge. This is a way to draw perfect circles and lines. Not all regular polygons can be made this way. A famous thinker named Gauss found the rules for these shapes.

Imagine a shape that is perfectly balanced in every way. In geometry, we call this a regular polygon. To be regular, a shape must follow two main rules. First, every side must be exactly the same length. Second, every corner, or angle, must be the same size.

These shapes can be simple and flat, which we call convex. They can also cross over themselves to look like stars.

These star shapes are known as regular star polygons.

Regular polygons have many special properties that work the same way every time. All the corners of these shapes sit perfectly on a single large circle. This is called a circumscribed circle. You can also fit a smaller circle inside the shape. This inner circle touches the exact middle of every side.

Because of these circles, regular polygons are very symmetrical. If you rotate the shape, it will look the same many times. This is called rotational symmetry. The number of times it looks the same matches the number of sides.

History shows us that people have studied these shapes for a long time. Ancient Greek mathematicians knew how to draw shapes with three, four, or five sides. They could also draw shapes with double the sides of ones they already knew. For a long time, people wondered if every regular shape could be drawn using only a compass and a straightedge. In 1796, a thinker named Carl Friedrich Gauss solved a big part of this puzzle. He proved that a regular 17-sided shape could be built this way.

There are many specific facts about how these shapes behave. A triangle has zero diagonals, but a square has two. As you add sides, the number of diagonals grows very quickly. For example, a pentagon has five diagonals.

If you add more and more sides, the shape starts to look like a circle. A shape with 10,000 sides is called a myriagon. Its internal angles are 179.964 degrees, which is almost a straight line.

Even with many sides, a polygon can never truly become a circle.

You can see these ideas in many places in our world. Even shapes that zig-zag through space can be regular. These are called regular skew polygons. They might look like a path winding between two flat planes.

Some even appear inside 3D shapes like a cube. You can also break even-sided polygons into smaller pieces. For instance, an eight-sided shape can be cut into several rhombi.

These patterns show how math connects simple lines to complex beauty.

{ "text": "In Euclidean geometry, a regular polygon is a shape defined by perfect balance. To be classified as regular, a polygon must be both equilateral and equiangular. This means every side has the exact same length, and every interior angle has the same measure.

These shapes are highly symmetrical. They possess rotational symmetry of order $n$, where $n$ is the number of sides. This means the shape looks identical to its original position $n$ times during a full rotation.

\n\nRegular polygons exhibit unique relationships with circles. All vertices of a regular polygon are concyclic, meaning they lie on a common circle called the circumscribed circle. Because the sides are equal, the shape also has an inscribed circle, or incircle. This inner circle is tangent to every side at its exact midpoint.

This dual relationship makes every regular polygon a tangential polygon as well. The distance from the center to a vertex is the circumradius ($R$), while the distance from the center to the midpoint of a side is the apothem ($a$).\n\nThere are different types of regular polygons based on their structure. The most common are regular convex polygons, which are simple shapes that do not intersect themselves. These can be identified by the Schläfli symbol $\{n\}$. For example, a square is $\{4\}$. There are also regular star polygons, which are non-convex and intersect themselves, such as the pentagram.

Beyond flat planes, there are regular skew polygons. These are nonplanar paths that zig-zag between two parallel planes, often seen in the edges of an antiprism.

Some even appear as Petrie polygons, which are paths of edges that divide a regular polytope into two equal halves.

\n\nHistory reveals a long quest to understand which polygons can be constructed using only a compass and a straightedge. Ancient Greek mathematicians could construct polygons with 3, 4, or 5 sides. They also knew how to double the number of sides of a known regular polygon. However, the question of whether all regular polygons were constructible remained a mystery for centuries. In 1796, Carl Friedrich Gauss made a massive breakthrough by proving the constructibility of the regular 17-gon.

Gauss later developed a theory showing that a regular $n$-gon is constructible if $n$ is the product of a power of 2 and any number of distinct Fermat primes. Pierre Wantzel later provided the full proof of necessity in 1837, a result now known as the Gauss–Wantzel theorem.\n\nAs the number of sides in a regular polygon increases, the shape undergoes a transformation. The interior angles grow larger, approaching 180 degrees. For a myriagon, which has 10,000 sides, the internal angle is 179.964 degrees.

While the shape begins to look like a circle as $n$ approaches infinity, it can never truly become one. If the angle reached exactly 180 degrees, the boundary would become a straight line, forming an apeirogon. This distinction is important: a circle is not a polygon with an infinite number of sides.\n\nMathematical properties also govern how these shapes can be divided. Any regular polygon with an even number of sides is a zonogon, meaning its opposite sides are parallel and equal in length. These can be dissected into several parallelograms. In the case of even-sided regular polygons, these parallelograms are specifically rhombi.

For instance, a 6-sided hexagon can be cut into 3 rhombi, while an 8-sided octagon can be divided into 6.

As the number of sides grows, the complexity of these dissections increases significantly.\n\nRegular polygons connect to many advanced areas of mathematics. They are used to study the properties of distances from arbitrary points in a plane to the vertices. For example, the sum of the perpendicular distances from any interior point to the sides of a regular $n$-gon is equal to $n$ times the apothem. This is a generalization of Viviani's theorem. These geometric principles provide the foundation for understanding more complex structures like uniform polyhedra and higher-dimensional polytopes.", "media": [ "File:PolygonParameters.png", "File:regular star polygons.svg", "File:annuli_with_same_area_around_unit_regular_polygons.svg", "File:Regular star figure 2(3,1).svg", "File:Antiprism17.jpg", "File:Petrie polygons.svg", "File:Polygons comparison.png", "File:8-gon rhombic dissection.svg", "File:6-gon rhombic dissection.svg" ] }