The area of regular polygons is used to measure the region covered by the polygon in two-dimensional space. Since it is a two-dimensional region, we use square units to measure the area. The area of a regular polygon can be calculated using the length of its apothem and the length of one of its sides. However, it is also possible to calculate the area of regular polygons by simply using the length of one of their sides.

Here, we will learn about the formulas that we can use to calculate the area of these polygons. Then, we will apply these formulas to solve some problems.

## Calculate the area of regular polygons using the apothem and sides

The area of any regular polygon can be calculated using the length of its apothem and the length of one of its sides. The area formula in these cases is:

$latex A=\frac{1}{2}nal$

where *a* is the length of the apothem, *l* is the length of one of the sides and *n* is the number of sides of the polygon.

This formula is derived from the fact that we can divide any regular polygon into triangles. For example, consider the following regular hexagon:

We can divide this hexagon into six congruent triangles. We know that the area of a triangle is equal to one-half of its base multiplied by its height. In this case, the height of the triangle is the apothem and the base is equal to one of the sides of the hexagon. Therefore, the area of each triangle is:

$latex A_{t}=\frac{1}{2}al$

Now, we see that we have six of these triangles (which is equal to the number of sides of the hexagon). Therefore, the area of the hexagon is:

$latex A=\frac{1}{2}(6)al$

$latex A=3al$

## Calculate the area of regular polygons using only the sides

The area of regular polygons can also be calculated using only the length of one of their sides. To achieve this, we can use the area formula that we saw earlier. However, we need to find an expression for the apothem in terms of its sides. An expression can be found using trigonometry.

We can use the tangent function to find the apothem. Therefore, the resulting formula is the following:

$latex A=\frac{{{a}^2}n}{4\tan(\frac{180}{n})}$

where *a* is the length of one of the sides of the polygon and *n* is the number of sides.

## Area of regular polygons – Examples with answers

The areas of the following regular polygons are found using the formulas seen above. Each example has its respective solution, where you can observe the process used.

**EXAMPLE 1**

If a pentagon has sides of length 8 m and an apothem of 5.5 m, what is its area?

##### Solution

A pentagon is a regular polygon with five sides. We use the area formula for regular polygons with the lengths $latex l = 8$ and $latex a = 5.5$. Therefore, we have:

$latex A=\frac{1}{2}nla$

$latex A=\frac{1}{2}(5)(8)(5.5)$

$latex A=110$

So the area of the pentagon is 110 m².

**EXAMPLE 2**

What is the area of a heptagon that has sides of length 8 m and an apothem of 8.3 m?

##### Solution

The heptagon is a seven-sided regular polygon, so we have $latex n = 7$. Furthermore, we have the lengths $latex l = 8$ and $latex a = 8.3$. Therefore, using the area formula with these values, we have:

$latex A=\frac{1}{2}nla$

$latex A=\frac{1}{2}(7)(8)(8.3)$

$latex A=232.4$

Thus, the area of the heptagon is 232.4 m².

**EXAMPLE 3**

Determine the area of a hexagon that has sides of 10 m.

##### Solution

The hexagon is a six-sided regular polygon, so we have $latex n = 6$. In this case, we only have the length $latex l = 10$. Therefore, using the second area formula:

$latex A=\frac{{{l}^2}n}{4\tan(\frac{180}{n})}$

$latex A=\frac{{{(10)}^2}(6)}{4\tan(\frac{180}{6})}$

$latex A=\frac{600}{4\tan(30)}$

$latex A=\frac{600}{2.31}$

$latex A=259.7$

Thus, the area of the hexagon is 259.7 m².

**EXAMPLE 4**

If an octagon has sides of length 5 m, what is its area?

##### Solution

The octagon is an eight-sided regular polygon, so we have $latex n = 8$. Similar to the previous exercise, we only have the length $latex l = 5$. Therefore, we use these values in the second formula for the area:

$latex A=\frac{{{l}^2}n}{4\tan(\frac{180}{n})}$

$latex A=\frac{{{(5)}^2}(8)}{4\tan(\frac{180}{8})}$

$latex A=\frac{200}{4\tan(22.5)}$

$latex A=\frac{200}{1.66}$

$latex A=120.5$

Thus, the area of the octagon is 120.5 m².

## Area of regular polygons – Practice problems

Apply the formulas for the volume of three-dimensional figures to solve the following practice problems. Select an answer and click “Check” to check that you got the correct answer.

## See also

Interested in learning more about geometric figures? Take a look at these pages:

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