Hexagons and their characteristics and uses are a central subject in geometry that regularly features in U.S. competitive examinations. A command of hexagons can be useful to those studying for the SAT (Scholastic Assessment Test), whereby questions on geometry regularly challenge understanding related to angles, symmetry, and the area. It can also assist with studying for tests like the ACT, GRE, GMAT, and AP Math tests, whereby polyhedron problem-solving is typical. This article discusses the kinds, characteristics, and uses of hexagons to prepare students with the competencies required to perform well in these exams.

What is Hexagon?

A hexagon is simply a kind of polygon that has six sides. The word "hex" in "hexagon" is a Greek word that means "six." Thus, a hexagon is defined as a polygon with six sides, six vertices, six interior angles, and six exterior angles. The sum total of the interior angles of a hexagon is 720°, and that of the exterior angles is 360°.

Have any of you ever closely observed the pattern in the honeycomb? You will find it very similar to hexagons. Actually, honey bees appear to have developed the capability of making perfectly hexagonal cells from the soft wax that they secrete as a by-product. But the question is: why hexagons? It’s basically a simple case of geometry. If you want to pack cells together that are similar in shape and size so that they fill all of a flat plane, only three regular possible shapes will work: squares, equilateral triangles, and hexagons. Of all these, the perimeter of the hexagon is least when compared with triangles or squares of the same area. So it's a good idea that honey bees would choose, perimeter of hexagons. Since making wax costs them lots of energy, they will want to use up as little as possible, similar to how builders want to save on the expenses of bricks. Isn't it amazing? Who taught this applied geometry to bees? Actually, this is a result of the concept of minimalism that nature follows.

Regular Hexagon 

Many times we confuse that a regular hexagon is just another name for a hexagon. But that is not actually true. There are two kinds of hexagons based upon the length of their sides: regular hexagons and irregular hexagons. A regular hexagon is a type of hexagon with six sides of equal length. If all six sides are equal, that implies all angles are also equal. Regular hexagons are also sometimes referred to as equilateral hexagons. Additionally, a regular hexagon is a convex geometrical shape. It implies that all the vertices of a regular hexagon will be pointing outside.

Similar to other regular polygons, regular hexagons also enjoy some kinds of symmetry. That means they have six lines of reflection symmetry and a rotational symmetry of order 6. Also, regular hexagons have equal internal angles of 120° each and central angles of 60° each.

Sides of a Hexagon

As we have earlier discussed, a hexagon is a plane, two-dimensional figure with six straight sides that enclose an area. The sum total of the lengths of all the sides of a hexagon gives the perimeter of a hexagon. In the case of regular hexagons, all sides are equal, thus,

Perimeter of hexagon = 6 × length of a side.

Similarly, if the perimeter of a regular hexagon is given, we can obtain the length of a side by

length of a single side = perimeter/6

Diagonals of a Hexagon

A diagonal is basically a line joining two non-adjacent corners or vertices of a polygon.

The formula to obtain the number of diagonals of any polygon can be given as,

No. of diagonals = [latex]\frac{n (n-3)}{2}

Here, n : number of sides of a polygon.[/latex]

For e.g., a hexagon has six sides, i.e., n = 6, thus,

No. of sides of a hexagon = 6(6-3)/2 = 9 diagonals

Hence, a hexagon has nine diagonals.

Types of Hexagon

As we have discussed in the introduction section, a hexagon can be classified into a regular hexagon, irregular hexagons, concave hexagons, and convex hexagons. Let's have a brief discussion on each type.

1. Regular Hexagon: As the name itself suggests, a regular hexagon has all sides equal in length and also has all its angles equal. Since all angles of a regular hexagon are equal, the measure of each interior angle of a regular hexagon can be given by 720°/6 or 120°.
2. Irregular hexagon: Just as the name implies, irregular hexagons are hexagons that are not regular. This means that the sides and angles of a hexagon are not equal, but other characteristics of hexagons will still be applicable to it. For example, it is still a two-dimensional closed polygon with six straight sides.
3. Concave Hexagon: A concave hexagon is a type of hexagon that has at least one interior angle greater than 180°. This hexagon has at least one corner or vertex that points inward.
4. Convex Hexagon: A convex hexagon is a type of hexagon that has all the interior angles measuring less than 180°. It has all its vertices, or corners, pointing outward, as the name implies. This is the type of hexagon that is very frequently used in geometry and its application.

Here is a diagram of all types of hexagon for your reference.

Properties of Hexagon

Now let's discuss some of the properties of hexagons:

• A hexagon is defined as a polygon with six sides, six vertices, six interior angles, and six exterior angles.
• It is a two-dimensional closed figure.
• The sum total of the interior angles of a hexagon is 720°.
• The sum total of all the exterior angles of a hexagon is 360°.
• The sum total of the lengths of all the sides of a hexagon gives the perimeter of the hexagon.
• In the case of a regular hexagon, the length of each side and the interior angles are equal to each other.
• The area of a regular hexagon is [latex]\frac{3√3a^2}{2} sq. units[/latex], where a is the side length.
• It has nine diagonals.
• Regular hexagons also enjoy some kinds of symmetry. That means they have six lines of reflection symmetry and a rotational symmetry of order 6.

Summary of Hexagon

Let's summarize the whole discussion so that you can revise quickly and efficiently.

• A hexagon is a six-sided polygon with six interior angles and six exterior angles. 
• It is basically a two-dimensional shape, which means it has only two dimensions: length and breadth, and no height.
•  A hexagon is defined as a polygon with six sides, six vertices, six interior angles, and six exterior angles.
• A diagonal is basically a line joining two non-adjacent corners or vertices of a polygon.
• The formula to obtain the number of diagonals of any polygon can be given as,

No. of diagonals = [latex]\frac{n (n-3)}{2}[/latex].

Solved Examples of Hexagon

Example 1. What will be the perimeter of a regular hexagon having side length 5 units?

Solution 1. Clearly, we know that, the perimeter of a regular hexagon is given by,

Perimeter = 6 × side length

[latex]\therefore[/latex] Perimeter = 6 × 5 units

= 30 units.

Example 2. What will be the area of a regular hexagon having side length 5 units?

Solution 2. Clearly, we know that, the area of a regular hexagon is given by,

Area = [latex]\frac{3√3a^2}{2} sq. units[/latex]

\therefore Perimeter = [latex]\frac{3√3×4^2}{2} sq. units[/latex]

= 24√3 sq. Units.

Example 3. Find the number of diagonals of a polygon having 8 sides?

Solution 3. Clearly, we know that the number of diagonals is given by,

Number of diagonals = [latex]\frac{n (n-3)}{2}[/latex].

= [latex]\frac{8 (8-3)}{2}[/latex]

= 20 diagonals. 

Grasping the categories, characteristics, and uses of hexagons is critical in succeeding in US competitive tests like the SAT, ACT, GRE, and AP Mathematics exams. Hexagons with their distinguishing symmetry, angle, and geometrical benefits show up quite regularly in problem-based questions. Domination of the said concepts, in addition to facilitating problem-solving techniques, sets up students in their ability to overcome intricate questions involving geometry effortlessly. By practicing and implementing these principles effectively, students can enhance their scores and overall performance in competitive exams. Just download the Testbook app and get started in your journey of preparation.