Pentagon Area Calculator

This pentagon area calculator can calculate the area of a regular pentagon in multiple different ways. You just have to enter one single measurement, either the side length, perimeter, diagonal, height, circumradius, inradius or area and our tool will automatically calculate all the others.

After entering a value and selecting the unit, the remaining values are automatically filled in, and the area is shown as an highlighted result. No prior setting is required since a regular pentagon is fully defined by a single measurement.

What is a pentagon?

A pentagon is a closed, flat geometric shape with five straight sides and five corners or vertices. The name comes from the Greek word for "five corners". A regular pentagon is one in which all five sides are of equal length and all five angles are of equal size. This tool calculates the area of such regular pentagons.

Pentagons that do not have all sides and angles equal are irregular pentagons, and their area must be calculated by other methods. If all the vertices point outwards, it is a convex pentagon; if at least one vertex points inwards, it is a concave pentagon. A regular pentagon is convex and has a high degree of symmetry with five axes of symmetry, so it can be fully defined by the length of one side.

Overview of pentagons.

Regardless of their size, all regular pentagons have the same angles and number of sides. These values are a direct result of all five sides being equal in length.

Formula for area of regular pentagon

The area of a regular pentagon is determined only by the length of its sides. The standard formula multiplies the square of the side length by a constant factor that involves the square root of five.

where A is the area and a is the length of one side. As this factor is approximately 1.720, the area of a regular pentagon is about one and three quarter times the square of the side length.

The inradius (or apothem) of a regular polygon is the distance from the center to the midpoint of one of the sides, and the circumradius is the distance from the center to one of the vertices.

Method with distance from center to side and five triangles.

To make the derivation of a formula easy to understand, it is best to draw a line from the middle to each corner point. This divides the pentagon into five identical isosceles triangles. The base of each triangle corresponds to the length of one side and the height corresponds to the distance from the center to the midpoint of the side.

If you add these values together, you get the value of half the perimeter multiplied by the distance from the center to the midpoint of a side. This is a general formula that works for any regular polygon.

This tool shows the area of each of those five triangles in a results map so you can see how your pentagon is broken down into these parts. If you know the distance from the center to the midpoint of one side, then you can directly calculate the area. That's why the distance from the center to the midpoint of one side is an input field.

Check all of the measurements for the pentagon at once.

Any measurement of a regular pentagon is related to any other measurement by the length of one side. The following table summarizes these relationships and explains why entering one value is sufficient to calculate the remaining values.

An example of a calculation.

Consider a regular pentagon with side length of 10 centimeters. Squaring the side length gives us 100, and multiplying that by the pentagon factor.

So the area of a pentagon with a side length of 10 centimeters is approximately 172.05 square centimeters. The perimeter of that same pentagon would be 50 centimeters, the diagonal would be about 16.18 centimeters, the height would be approximately 15.39 centimeters, the radius of the circumscribed circle would be around 8.51 centimeters and the distance to the center from a side would be about 6.88 centimeters.

If you use this calculator, all you have to do is enter the value 10 into the field for side length to get all of these values instantly. You can also start with another measurement and in that case the tool will show you the side length.

Examples of areas for common side lengths:

The following table shows some examples of areas for common side lengths. The results are rounded to two decimal places. Note that the area increases rapidly as the length of the sides gets larger since it is squared when calculating the area.

The golden ratio and the five-pointed star.

The pentagon holds one of the most famous numbers in geometry. If you draw five diagonals they cross each other and form a five-pointed star, also called a pentagram. The ratio between the lengths of the diagonals and sides is exactly equal to the golden section, which is denoted by the Greek letter phi.

This fact affects the whole shape. Each diagonal is divided by other diagonals into several line segments, and these segments are also in a ratio of the golden section. In the center of the five-pointed star there is a smaller regular pentagon turned upside down. The length of its sides corresponds to the value of the original side length divided by the square of the golden section.

Over the centuries, the pentagram has remained a controversial symbol. The ancient Pythagoreans used it as an identifying mark and later it appeared on flags, coats of arms and seals. If you enable the golden ratio option, you can see the inner pentagon and diagonals in your shape.

Pentagon and Circle

Any regular pentagon can be inscribed in two circles of different radii. The larger circle passes through the five vertices, and its radius is the circumradius. The smaller circle touches each side at its midpoint, and its radius is the apothem.

Because the shape is constant, the area of the pentagon will always be about 75.7% of the larger circle and about 15.6% more than the smaller circle. These ratios do not change with size. Because the pentagon has only five sides, there is more empty space inside the circle than an octagon would have. So a pentagon does not look as round as it looks angular.

Pentagon and dodecahedron

The regular pentagon is one of the five Platonic solids, being the face of a dodecahedron. The dodecahedron consists of twelve identical pentagons. The Greeks considered the dodecahedron to be the shape of the universe itself and saw it as a fifth element beyond earth, water, air and fire.

Thus the d12 die, which is a Dodecahedron, has a pentagon on each of its faces. The pentagon can also be found in other Archimedean Solids, including the truncated Icosahedron. The truncated icosahedron is known for its black and white pattern often seen on soccer balls, and consists of twelve black pentagons and twenty white hexagons arranged alternately.

Is it possible to construct a pentagon with compass and ruler?

Yes, and this is also one of the differences between a pentagon and many other related geometric shapes. With only a compass and an unmarked straightedge, a regular pentagon can be accurately drawn. This method has been traced back to Euclid's "Elements" over two thousand years ago.

The reason for this is that five is a special prime number, a Fermat prime, and this is the condition under which a regular polygon can be constructed with compass and straightedge. While a regular heptagon (seven sides) or nonagon (nine sides) cannot be constructed in this way, pentagons, squares and equilateral triangles can be. The construction begins by establishing the golden ratio, which is then used to determine each vertex in turn.

Where do pentagons appear?

The most famous pentagon in the world is the Pentagon, headquarters of the US Department of Defense. The building derives its name from the flat arrangement of a pentagon. Each exterior wall measures about 921 feet long and it is one of the largest office buildings in the world with an almost perfect regular pentagonal shape.

In sports the home base on baseball fields is pentagonal and the black panels of traditional soccer balls are regular pentagons. While fivefold symmetry is rare among minerals, it is common for living things, which is why you find pentagons in flowers with five petals, in cross-sections of okra pods, in a particular arrangement in apple cores, and in the five arms of sea stars.

Pentagons are also found in the geometric structure of coins and medals as well as bridges and domes. Because of their simple outline that contains both the golden ratio and a pentagram they are often used as examples for teaching materials.

Tips for accurate results:

Before you can input a side length, make sure that your figure is actually a regular pentagon. The formulas assume that all five sides and angles are equal. Be careful to match up the units. If the length of one side is measured in centimeters, then the area will be calculated in square centimeters.

Since area changes according to the square of the side length, a small error in the side length will lead to a larger error in the area. If possible, measure one of the following values (diagonal, height, distance from center to side) as a reference and enter it here to ensure that the result is for the same pentagon.