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# Area of Hexagon – Formula, Definition, Examples

A hexagon is a six-sided polygon. It has six straight sides and six angles. The word “hexagon” comes from the Greek words “hex” meaning “six” and “gonia” meaning “angle.”

In nature, hexagons can be found in beehives, snowflakes, and some crystal structures. In construction, hexagonal shapes are sometimes used for tiles or paving stones to create unique patterns.

## What is the Area of a Hexagon?

The area of a hexagon refers to the amount of space enclosed by a hexagon, which is a six-sided polygon. To find the area of a hexagon, you need to know the length of its sides or its apothem (the distance from the center of the hexagon to its midpoint). According to the length of the sides, the hexagon can be of two types,

1. A hexagon can be either regular or irregular. A regular hexagon has six equal sides and six equal angles.
2. An irregular hexagon has sides and angles of different lengths and measurements.

## Area of Hexagon Formula

The formula to calculate the area of a regular hexagon (a hexagon with equal sides and angles) is:

A = (3√3/2) x s2

Where: A = area of the hexagon s = length of the side of the hexagon

## How to Find Area of Hexagon?

1. Understand the basics of a hexagon
2. Identify the length of the side of the hexagon
3. Use the formula to calculate the area of a regular hexagon: The formula to calculate the area of a regular hexagon is: A = (3√3/2) x s2

Where: A = area of the hexagon s = length of the side of the hexagon

To use this formula, simply square the length of the side of the hexagon, multiply it by 3√3/2, and you’ll have the area of the hexagon.

1. Plug in the values: Once you have identified the length of the side of the hexagon, plug in the value of s into the formula and solve for A.

For example, let’s say the length of the side of the hexagon is 5 cm. To find the area of the hexagon, we would use the formula:

A = (3√3/2) x 5^2 A = (3√3/2) x 25 A = 32.48 cm^2 (rounded to two decimal places)

So, the area of the hexagon with a side length of 5 cm is approximately 32.48 square centimeters.

1. Use a different formula for irregular hexagons: If you have an irregular hexagon with sides and angles of different lengths and measurements, you’ll need to use a different formula to find its area. One way to do this is to divide the irregular hexagon into smaller shapes that you can find the area of, such as triangles or trapezoids, and then add them up to find the total area of the hexagon.

## Solved Examples on Area of Hexagon

Example 1: Find the area of a regular hexagon with a side length of 6 cm.

Solution: A regular hexagon is a polygon with six congruent sides and angles.

The formula to find the area of a regular hexagon is:

Area = (3√3 / 2) x s²

where s is the length of a side.

Substituting the given value of s = 6 cm in the formula, we get:

Area = (3√3 / 2) x 6² = (3√3 / 2) x 36 = 54√3 cm²

Therefore, the area of the regular hexagon is 54√3 cm².

Example 2: A regular hexagon has an apothem of 5 cm and a side length of 7 cm. Find its area.

Solution: An apothem of a regular polygon is the distance from the center of the polygon to the midpoint of a side. The formula to find the area of a regular hexagon is:

Area = (3√3 / 2) x s²

where s is the length of a side.

However, we need to first find the length of the radius (r) of the circumscribed circle, which is given by:

r = apothem / cos(30°) (since the interior angle of a regular hexagon is 120°, and half of it is 60°, which is the angle between the apothem and a radius)

Substituting the given values of apothem = 5 cm and cos(30°) = √3 / 2, we get:

r = 5 / (√3 / 2) = 10 / √3

Now, we can find the length of a side (s) using the Pythagorean theorem, since the radius, apothem, and one of the sides form a right triangle:

s² = r² – apothem² = (10 / √3)² – 5² = 100 / 3 – 25 = 75 / 3 = 25

Therefore, the length of a side is s = √25 = 5 cm.

Finally, substituting the values of s and using the area formula, we get:

Area = (3√3 / 2) x 5² = (3√3 / 2) x 25 = 37.5√3 cm²

Therefore, the area of the regular hexagon is 37.5√3 cm².

## Frequently Asked Questions on Area of Hexagon

Here are some frequently asked questions on the area of hexagons:

Q: What is a hexagon?

A: A hexagon is a six-sided polygon.

Q: What is the formula for finding the area of a regular hexagon?

A: The formula for finding the area of a regular hexagon is:

Area = (3√3 / 2) x s²

where s is the length of a side.

Q: What is the difference between a regular and irregular hexagon?

A: A regular hexagon has six congruent sides and angles, while an irregular hexagon has sides and angles of different lengths and measures.

Q: Can you find the area of an irregular hexagon?

A: Yes, you can find the area of an irregular hexagon by dividing it into smaller shapes whose areas can be calculated and then summing them up.

##### Q: How do you find the apothem of a regular hexagon?

A: The apothem of a regular hexagon is the distance from the center of the polygon to the midpoint of a side. It can be found using the formula:

apothem = s / (2 x tan(180° / n))

where s is the length of a side, and n is the number of sides (in this case, n = 6).

Q: What is the maximum area a hexagon can have?

A: The maximum area a hexagon can have is achieved by a regular hexagon, since it has the most symmetry and regularity among hexagons.