Quick navigation:
- How to use the hexagon calculator
- Hexagon formulas
- Area of a hexagon formula
- Perimeter of a hexagon formula
- Hexagon diagonals formulas
- Equations for hexagon radiuses
How to use the hexagon calculator
This calculator can completely solve a regular hexagon (a.k.a. a six-sided polygon) by entering just one of these parameters:
- the side length (a) of any side a all six sides have equal length
- the diagonal (d), all three diagonals being of the same length
- the short diagonal (s), all of them being of equal length
- the radius (R) length, a.k.a. circumradius - the radius of the circumcircle touching all six hexagon vertices
- the apothem (r), a.k.a. the inradius - the radius of the incircle which can fit exactly inside the hexagon
- the hexagon perimeter (P)
- the hexagon area (A)
Entering one value means all of the others in the list can be calculated from it. This means you can use it as a hexagon area calculator, hexagon diameter calculator, hexagon side length calculator, etc.
To use the tool, first select what is known about the hexagon, then enter the value in the right unit, and finally choose which units the output should be in. Supported units include mm, cm, dm, meters, km, inches (in), feet (ft), yards (yd), and miles, as well as their squares for the area (sq cm, sq in, etc.) with conversion done automatically.
Hexagon formulas
All formulas for solving a regular hexagon shown use the notation as shown in this graph:
All formulas assume the side of the hexagon is known so the first step in applying a formula depending on the starting point is to calculate the side length by what is given as input. E.g., to find the side from the perimeter, simply divide the perimeter by six.
No formula is given for finding a hexagon angle since in all regular hexagons the angles are all equal to 120°. If one draws the long diagonals they all split these exactly into two, forming six equilateral triangles (equal-sided triangles) which have all three angles being 60°.
Area of a hexagon formula
To find the area of a regular hexagon use the following equation:
A = 3/2 · √3 · a2
where a is the length of the hexagon side. If the area is known and the side is not, the formula can be reversed to find the side as such: a = √(A / (3/2 · √3)).
A second, more generic formula for the area of any regular polygon, is:
A = P · r / 2
where P is the perimeter of the polygon and r is its apothem (incircle radius).
Perimeter of a hexagon formula
To find the perimeter of a hexagon, simply multiply its side by six:
P = a · 6
To find the side from the perimeter, divide the perimeter by six.
Hexagon diagonals formulas
The long diagonal (d) is just two times the side: d = a · 2, hence the side is half of the diagonal leading to the equation a = d / 2.
The short diagonal (s) can be calculated using the formula: s = a · √3, conversely a = s / √3.
Equations for hexagon radiuses
The circumradius (R) is simply equal to the side, so if the side length is known, then the radius is also known.
To find the inradius (r) of a hexagon use the formula: r = a · √3 / 2, conversely a = r * 2 / √3.
Circumradius and inradius of a hexagon
Some might be confused about the fact that hexagons have not just one, but two radiuses. These are, in fact, the radiuses of the circles which contain all vertices and all central points on the sides of the hexagon, and are often important in solving hexagon-related geometry tasks.
The circumradius is the radius of the circumference that contains all vertices of a hexagon. In a regular hexagon, its length is exactly equal to the length of its side (R = a).
The inradius is the radius of the largest circle which can be contained in its entirety within the hexagon and is also the hexagon's apothem. The apothem is the distance between the center of the hexagon and the midpoint of any side, which always forms a right angle.
Example hexagon geometry problems
Example 1: Find the area of a hexagon given its perimeter is 12 cm.
Solution: This can be approached most easily as a two-step task. First, find the side of the hexagon by knowing that P = 6 · a, hence a = P / 6. So the side length is 12 / 6 = 2 cm. Then, use the hexagon area equation and calculate 2/3 · √3 · 2 = 0.866 cm2 (square centimeters).
Example 2: Find the area of a hexagon in square meters if it is known that it has a long diagonal equal to 10 feet.
Solution: To find the area via the diagonal, decompose the task into two steps. First, we find the length of the side of the hexagon which is simply the diagonal divided by two. So a = 10 / 2 = 5 feet. Then, using the area formula we find that 2/3 · √3 · 5 = 16.2380 ft2 (sq ft). Finally, convert square feet to square meters to arrive at the answer of 1.5086 m2 or roughly one and a half square meters.
Example 3: What is the area of a hexagon with side 1?
Answer: This task can be resolved directly using the hexagon area formula. If the side length is one inch, to find the area, multiply 2/3 by √3 and then multiply by 1: 2/3 · √3 · 1 = sqrt(3) / 4 = 0.433 square inches since the square root of 3 is 1.732051.
Practical application of hexagons
Six-sided polygons and primarily regular hexagons see a lot of applications in both nature and man-made objects. Everyone is familiar with the honeycomb pattern of beehive structures. Less familiar are the eyes of insects which typically consist of many six-sided light receptors bunched together. Photography lenses as well as the lenses of telescopes used in astronomy also mirror that design and consist of a number of hexagons. A notable example is the James Webb space telescope.
A few reasons explain the popularity of the hexagon shape. First, regular 6-sided hexagons have the smallest perimeter per unit of area among surface-filling polygons meaning less material is needed to construct a structure or element to span a certain area. Furthermore, the 120° angles distribute forces (mechanical and tension stress) amongst their adjacent sides equally making hexagonal constructs stable and efficient. This is why a hexagon calculator is often used in engineering applications. Finally, a number of regular hexagons can fill a surface with no gaps between them, which is a property shared with just a few other shapes which include regular triangles and squares.
