Hexagon Area Calculator

This hexagon area calculator allows you to calculate the area of a regular hexagon in multiple different ways - using many different inputs. Whether you know the side length, perimeter, long or short diagonal, circumradius, distance from center to the midpoint of a side, or even just the area itself - this tool will do all the work for you by simply entering one of these values.

After entering a value and selecting the unit, the remaining values will be automatically displayed with the area being highlighted as the result. Since a regular hexagon is already defined by a single measurement, you do not need to make any adjustments.

What is a Hexagon?

A hexagon is a closed, flat shape with six straight sides, six corners and six interior angles. The name comes from the Greek words for "six" and "corner".

A regular hexagon is a shape with six sides where all of the sides are equal in length and all of the angles are the same. This tool will only calculate the area of the regular shapes strictly. Irregular hexagons, with sides of different lengths and/or angles of different measure need to be broken down into other shapes such as triangles and then added together to find the total area.

A regular hexagon is also a convex polygon, meaning that every corner points outward. It is also a highly symmetrical figure with six axes of symmetry and six-fold rotational symmetry. This symmetry allows the entire shape to be defined by just one measurement.

A look at some facts about hexagons

Regardless of size, all regular hexagons have the same angles and number of sides. These values can be derived directly from the six equal length sides.

Formula for area of a hexagon

The area of a regular hexagon is determined only by the length of one of its sides, and in the standard formula this length is squared and multiplied by a constant.

In this formula A is the area and a is the length of one side. As that constant is approximately 2.598, it can be said that for regular hexagons the approximate area is equal to the product of the side's length multiplied by itself and then multiplied by 2.598.

Even if the length of the sides is not known but instead the distance from the center to a side or the radius of the circumscribed circle are given, one can calculate the area. The distance r from the center to a midpoint of a side and the radius R of the circumscribed circle to the center of a vertex.

It is no coincidence that the area calculated from the radius of the circumcircle matches exactly with the area calculated from the length of the sides. In a regular hexagon, the radius of the circumcircle and the side length are equal.

Six equilateral triangles

To really understand the formula for finding the area of a hexagon, it is helpful to draw three long diagonals and divide the hexagon into six equal triangles. The sides of each of these triangles are the same length as the sides of the hexagon.

The area of an equilateral triangle with side length a is the square of a multiplied by the root of 3 divided by 4. The area of the hexagon results from adding up the areas of these six triangles.

This tool shows the area of each of the six triangles on a results map so you can see how your hexagon is broken down into those parts.

A method with the distance from center to a side.

There is another way to calculate the area of regular polygons which works for all types. Drawing lines from the center point to each vertex divides the hexagon into six triangles. The base of each triangle corresponds to a side of the hexagon, and the height corresponds to the distance from the center of the sides.

Adding these values together gives the value that is created by multiplying half of the circumference length with the distance to the center of the sides, which results in the general formula for regular polygons.

This is why the distance to center of sides is so useful. If you can measure the distance from the center to a side's midpoint, then you can directly calculate the area. For this reason, this tool accepts the distance to center of sides as an input value.

Check all values at one point

Each measurement of a regular hexagon is related to any other measurement by the length of its sides. The following table shows the relationships that this tool uses so that when one value is entered, the others can be calculated.

Example calculation

Consider a regular hexagon with side length of 10 centimeters. The square of the side length is 100, and then multiplied by a constant for the hexagon.

Thus the area of a hexagon with side length of 10 centimeters is approximately 259.81 square centimeters. The perimeter of this same hexagon would be 60 centimeters, its long diagonal would be 20 centimeters, its short diagonal would be about 17.32 centimeters and the distance from the center to a side would be about 8.66 centimeters.

If you use this tool, all you have to do is type the value 10 into the side length input box to get those values directly. You can also start with any other 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, rounded to two decimal places. As the side length increases, the area increases rapidly as the side length is squared.

Diagonals of a hexagon.

Diagonals connect two non-adjacent vertices. A hexagon has nine diagonals, which can be confirmed by the formula for polygons: multiply the number of sides by a number three less than itself, and then divide that product in half.

There are two different lengths of these nine diagonals. Three long diagonals run directly through the center from one corner to the opposite corner, each length being exactly twice the side length. Six short diagonals span across a corner, each length being the product of the side length and the square root of 3.

The long diagonals are the distances between two opposite vertices while the short diagonals represent the width of the flat, i.e. the distance between two opposite sides. Both are important input parameters for this tool so that the hexagon can be determined directly from a measured diagonal.

Radius equal to side length

Some geometric shapes have surprising properties. In the case of a hexagon, this property is the radius of its circumscribed circle. The radius of the circumscribed circle is the distance from the center to any vertex and also the radius of a circle that passes through all vertices. For a regular hexagon, this value is exactly equal to the side length.

This is made up of six equilateral triangles. Each triangle has the center as a common point with two neighboring vertices. As it is an equilateral triangle, the distance from the center to one vertex is equal to the length of the side. The radius of the inscribed circle is shorter and corresponds to the product of the length of the side and the square root of 3 divided by 2.

If you open the compare option, you can see two circles corresponding to the hexagon. One runs through each corner and one is inside each side.

The Hexagon and its Circles

Any regular hexagon will fit snugly between two circles. The larger circle passes through the six vertices, and its radius is the circumradius of the hexagon. The smaller circle touches the midpoint of each side, and its radius is the inradius of the hexagon.

As the shape is fixed, the area of the hexagon always takes up about 82.7 percent of the larger circle and is about 10.3 percent larger than the smaller circle. These proportions do not change regardless of size. This is a simple way to check that although the hexagon approximates the circle quite well, it still leaves slightly more space at the corners compared to denser polygons.

Where do you find hexagons?

The reason hexagons are found everywhere in nature and engineering is almost the same: they can fill a plane without gaps, which minimizes the length of walls required. A well-known example is the honeycomb built by bees. This arrangement is so efficient that it's known as "the honeycomb principle."

Hexagons can be found in the basalt columns of Giant's Causeway, snowflakes, the strange permanent cloud patterns at Saturn's north pole and the lattice structure of graphene, a material that is just one atom thick. The James Webb Space Telescope uses eighteen hexagonal mirrors with sides about 0.737 meters long. Hexagons can be packed together to form a shape similar to a large round mirror field.

In everyday life, hexagons are used in bolts and nuts, pencils that don't roll off desks, tiles for floors and bathrooms, as well as the fields of board games. The spacing measurements of the sides of a nut is the short diagonal which can be determined with this tool making it also a quick reference to determine the size of a wrench.

Why hexagons are everywhere.

Some properties explain why this shape is so popular. Of all polygons that can completely tile a plane, the regular hexagon has the least perimeter for a given area. So hexagonal structures can cover an area with minimal material.

The 120-degree angle ensures that mechanical stresses and tensile forces are evenly distributed between neighboring walls, making hexagonal frames stable and resilient. There are only three regular polygons that can fill a plane by themselves: triangles, squares, and hexagons. Of these, hexagons form the largest cells for a given wall length.

Tips for accurate results.

Before using the length of a side, make sure that your figure is actually a regular hexagon. The formula requires that all six sides and internal angles be the same. Be careful to keep units consistent. If you measure the length in centimeters, then the area will be in square centimeters.

Note that small errors in the side length can lead to large errors in the calculated area since the area is proportional to the square of the side length. If possible measure the distance from center to a side or diagonal as a reference and enter it here to ensure you get the result for the same hexagon.