Pentagon

A pentagon is a shape with five sides.

You can make one with a strip of paper. Just tie a knot and flatten it out.

A regular pentagon has sides that are all the same. Its five corners also look the same.

Sometimes, the lines cross to make a star. This is called a pentagram.

It is fun to look for these shapes!

A pentagon is a shape with five sides.

Most people know the regular pentagon. In this shape, all five sides are the same length. All five corners also look the same. Each corner has an angle of 108 degrees.

You can find symmetry in this shape. Symmetry means it looks the same after you flip or turn it. A regular pentagon has five lines of reflection. This means you can flip it in five ways. It also has rotational symmetry. You can turn it five times to see the same view.

Sometimes, the lines of a pentagon cross each other. This makes a star shape. We call this a pentagram.

You can even make a pentagon with paper. Tie a knot in a strip of paper. Then, pull the ends to flatten it. This makes a pentagon shape.

Regular pentagons are tricky to use for tiling. Tiling means covering a floor with shapes. You cannot cover a flat floor using only regular pentagons. They will always leave gaps or overlap. This is because their angles do not fit together perfectly.

A pentagon is a special shape in geometry. It is any polygon that has exactly five sides.

Some pentagons are simple, which means their lines do not cross. Other pentagons are self-intersecting. If a regular pentagon has lines that cross, it creates a star shape. This star is called a pentagram.

You might see these shapes in art or nature. They are interesting because of how their sides and angles work together.

In a regular pentagon, everything is perfectly balanced. All five sides are the same length. All five interior angles are exactly 108 degrees.

Because it is so balanced, it has many symmetries. It has five lines of reflection where you can flip it. It also has rotational symmetry of order 5. This means you can turn it by 72 degrees and it looks the same. You can also turn it by 144, 216, or 288 degrees.

People have studied how to draw these shapes for a long time. Around 300 BC, a mathematician named Euclid described how to make one.

He used a compass and a straightedge to do this. There are other ways to build them too. One method uses something called Carlyle circles. Another way uses trigonometry to find the right lengths. Even a simple strip of paper can make one if you tie an overhand knot.

There are many math facts about the pentagon's size. The sum of all its inside angles is 540 degrees. If you draw a circle that touches all five corners, it is called a circumscribed circle.

A regular pentagon fills about 0.7568 of that circle. You can also draw a smaller circle inside the shape. This is called an inscribed circle. The distance from the center to the side is called the apothem.

These numbers help mathematicians calculate the exact area of the shape.

Pentagons are very useful for understanding how shapes fit together. Some pentagons can tile a plane, which means covering a surface without gaps. However, regular pentagons cannot do this.

If you try to put them together, they will always leave gaps. This happens because their 108-degree angles do not add up to a full circle. Scientists have found that the best way to pack them is a double lattice structure.

This pattern covers about 92.131% of a flat surface.

In geometry, a pentagon is any polygon with exactly five sides. This shape is a fundamental building block in the study of polygons. A pentagon can be simple, meaning its sides do not cross each other. It can also be self-intersecting. When a regular pentagon is self-intersecting, it forms a star shape known as a pentagram

. The properties of these shapes depend on how their sides and angles are arranged.

A regular pentagon is a highly balanced version of the shape. In a regular pentagon, all five sides are equal in length. All five interior angles are also identical, measuring exactly 108 degrees. The sum of all internal angles in a simple pentagon is always 540 degrees. Because of this perfect balance, regular pentagons possess significant symmetry

. They have five lines of reflectional symmetry. They also have rotational symmetry of order 5. This means the shape looks the same after rotations of 72, 144, 216, or 288 degrees.

Mathematicians use specific measurements to describe the size of a regular pentagon. Every regular convex pentagon has an inscribed circle, which sits inside the shape. The radius of this circle is called the apothem. The pentagon also has a circumscribed circle that passes through all five vertices

. A regular pentagon fills approximately 0.7568 of the area of its circumscribed circle. The diagonals of a convex regular pentagon exist in the golden ratio to its sides. This mathematical relationship connects the pentagon to other complex geometric concepts.

Constructing a regular pentagon is a classic problem in geometry. Since 5 is a Fermat prime, the shape is constructible using only a compass and a straightedge. This method was famously described by Euclid in his work, Elements, around 300 BC

. Other sophisticated methods exist for creating the shape. Richmond's method uses a unit radius circle and specific angle bisections to find the side length. The Carlyle circle method uses circles to find the roots of quadratic equations to locate vertices. There is also a method using trigonometry that divides the pentagon into ten congruent triangles

. Even a simple strip of paper can form a pentagon through an overhand knot.

There are different types of pentagons beyond the regular version. An equilateral pentagon has five sides of equal length, but its angles can vary. A cyclic pentagon is a shape where all five vertices lie on a single circle. A special type of cyclic pentagon is called a Robbins pentagon. These are unique because they have both rational sides and a rational area. There are also 15 distinct classes of pentagons that can tile a plane. These are shapes that can cover a flat surface without leaving any gaps.

One of the most interesting facts about regular pentagons is that they cannot tile a plane. To tile a plane perfectly, the angles meeting at a vertex must add up to 360 degrees. Because a regular pentagon's angle is 108 degrees, they cannot fit together without gaps. For example, three pentagons leave a gap, while four would overlap. The best-known way to arrange them is a double lattice structure

. This specific packing covers 92.131% of a plane. In 2016, researchers Thomas Hales and Wöden Kusner announced a proof regarding this optimal density.

Pentagons relate to many broader mathematical and physical systems. They appear in the study of polyhedra and complex symmetry groups. For instance, the symmetry of a regular pentagon is described as Dih5 symmetry, which has an order of 10. They also relate to the study of tiling and packing efficiency in mathematics. Understanding the pentagon helps scientists and mathematicians model how objects occupy space. From simple paper knots to complex geometric proofs, the pentagon remains a central subject of study.