Pentagon Perimeter Calculator

This pentagon perimeter calculator allows you to find the perimeter of a regular pentagon, which is a five-sided polygon with sides having equal length and internal angles with equal measure. It also offers more functions such as calculating all the other values once one known value (side length, perimeter, distance from center to an edge, circumscribed circle radius or area) is given. By entering a number and selecting its unit you will automatically have the rest of fields filled in, showing prominently the pentagon perimeter. The pentagon is the base shape for the pentagram, and each of its measurements are determined by the famous golden ratio. This tool allows you to verify this with one click.

What is the perimeter of a pentagon?

A pentagon is a polygon with five sides and five corners. When all five sides are equal in length and all five inside angles are the same, it is called a regular pentagon. This is the shape that this tool deals with. The perimeter is the total length of the boundary, so the distance along the outside edge.

For a regular pentagon, the perimeter is equal to five times the side length. It can be measured using common units of length such as centimeters, meters or inches.

How to use this calculator:

Enter a known measurement in the appropriate field and select the unit (e.g., centimeters or inches). After entering, the tool will automatically calculate the remaining four values and prominently display the circumference. It is not necessary to specify which value was originally entered.

If you want to repeat the calculation with another known value, delete the previously entered fields and enter the new value. By opening the "Golden Ratio" option, you can see a pentagram and a pentagon nested inside it. By opening the comparison option, you can compare a pentagon with its circumscribed circle and inscribed circle. By opening the diagram, you can observe how the perimeter changes as the side length increases.

Formula for the perimeter of a pentagon

The perimeter of a regular pentagon is determined only by the length of its sides. As there are five equal-length sides, we multiply the side length by 5.

In this formula, p is the perimeter and a is the length of one side. Let's say that each side has a length of 10 cm. We would have:

Conversely if you know the perimeter and want to find the length of a side, divide it by five.

This is the same rule that applies to any regular polygon, with only the number five replaced by the number of sides. This simple formula works only when all sides are equal in length.

Calculation based on side center, circumcircle radius or area.

It is not necessary to start with the side length. The center of a side is the distance from the center point to the midpoint of one side, the circumradius is the distance from the center point to a vertex and the area describes the space enclosed by the figure. All these values are geometrically connected and depend on the side length since they are determined by the five triangles that make up the pentagon. Therefore, if any of these values is known, one can calculate back to the side length and from there determine the perimeter.

In these formulas r is the inradius, R is the circumradius and A is the area. The angles are exactly what you get when you divide 180 by 5, leading to a nice value of 36 degrees. Since a regular pentagon can be constructed with compass and straightedge, these constants can be expressed in an exact form involving the square root of five. This same square root is hidden in the golden ratio. The trigonometric calculations can be left to a calculator; you just need to enter the values that are given.

Check all measurement values of the pentagon.

Any measurement of a regular pentagon is related to the others by the side length. The following table shows the relationships used by the calculator. So if one of these values is entered, all the others can be calculated.

Example calculation.

The following table shows the perimeter for various typical side lengths. Note that as the side length increases, so does the perimeter, and they are both directly proportional because the side length is not squared.

It is also quick to do the opposite starting from the perimeter. For a regular pentagon with a perimeter of 60 centimeters, the side length would be 12 centimeters which is the result of dividing 60 by 5. If you enter 60 in the "Perimeter" column, the side length will immediately appear as well as the distance from center to vertex, the radius of the circumscribed circle, the diagonal and area.

Diagonal and golden section

A diagonal is a line segment that connects two non-adjacent vertices. A pentagon has five diagonals, and in a regular pentagon they all have the same length. This length is one of the most famous puzzles associated with pentagons, as it is equal to the product of the side length and the golden ratio.

If you draw five diagonals inside a regular pentagon they will intersect and form a pentagram (a five-pointed star). Each time two of the diagonals cross, one of them is divided exactly in the golden ratio, which is the ratio between the longer part and the shorter part. In the center of the star there is a smaller regular pentagon whose side length is the result of dividing the original side length by the square of the golden ratio, which is approximately 0.382 times the original side length. You can continue this process inside that smaller pentagon, drawing ever smaller pentagrams. This pattern repeats infinitely, with each pentagon being a factor of the golden ratio smaller than the previous one.

Two simple relationships emerge from these observations that apply to all pentagons. Subtracting the diagonal length from the side length yields a value equal to the side length divided by the golden ratio. The square of the diagonal is equal to the sum of the square of the side length and the product of the side length and the diagonal, which is exactly the equation that defines the golden ratio. If you enable the golden ratio option, you can see the calculated diagonal and nested pentagons based on your input values.

Angle of a regular pentagon

The angles of a regular pentagon are constant and produce whole number results no matter how large or small the figure is drawn. Each of its five interior angles measures exactly 108 degrees, and their sum is 540 degrees. The central angle formed by connecting the center point to two adjacent vertices also measures exactly 72 degrees, as do the exterior angles.

Why is it possible to construct a pentagon with compass and ruler?

A regular pentagon is one of the figures that can only be drawn precisely with a compass and an unmarked ruler. This fact has been known since ancient Greek times and was described in Euclid's "Elements". Not all polygons can be constructed in this way. For example, a regular heptagon or a regular nonagon are not possible, while a pentagon is. This is because five is a special prime number that is known as a Fermat prime.

This explains why the golden ratio appears in the measurements of a pentagon. Figures that can be constructed with compass and ruler are representable by normal square roots. The golden ratio is made up of the square root of five. Therefore, there are simple and precise formulas for diagonals, distances to center from a side, circumradius and area. In contrast, the constants for heptagons and nonagons are trigonometric values of irrational numbers and cannot be brought into a simple square root form.

Regular and irregular pentagons.

This tool is for a regular pentagon, which means that all five sides are the same length. If your pentagon is irregular, meaning the sides are not all the same length, then you cannot use this simple method of multiplying one side by 5 to find the perimeter. In this case, you must measure each side individually and add up all the lengths to get the total perimeter. A baseball home plate is an example of an irregular pentagon, as it has a square base with a triangular point. Therefore, not all five sides are the same length. The formulas for the distance from the center to a vertex, circumradius, diagonal, and area only apply to regular pentagons.

Using the circumference of a pentagon:

If you look closely enough, pentagons are everywhere. The U.S. Department of Defense conducts its business inside the Pentagon, which has a huge regular pentagon as an outer wall. A soccer ball is made up of twelve black pentagons and twenty white hexagons sewn together. The home plate in baseball is a pentagon, and the pentagram (five-pointed star) appears on countless flags and coats of arms, including the stars on the U.S. flag.

If the perimeter is known, this information allows a manufacturer to calculate how much material will be required to install borders, trim, fencing or frames around the perimeter of the pentagon. Architects and builders use the same relationship to fit a pentagon into a circle of a given size, or determine the size of a circle that would exactly enclose a pentagon.

Tips for accurate results:

Carefully measure each side and let a calculator do the multiplying rather than measuring all the way around. Make sure that all units are the same. If you input the length of the sides in centimeters, then the perimeter will also be in centimeters. Since formulas for regular pentagons assume that all five sides are equal in length, check to see if it is a regular pentagon before using methods to find the distance from the center to the side, the radius of the circumscribed circle, the diagonal or area.