What is the formula for finding the area of a regular polygon with perimeter P and apothem length a

You have one formula, $A=\frac12 aP$, that is generally true of any regular polygon, and moreover you have the formula $A = P$ which is true of a particular regular polygon mentioned in the problem statement.

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The fact that $A = P$ means that you can use $A$ or $P$ interchangeably in any formula describing this particular polygon: $A$ and $P$ are the same number. So the formula $$A=\frac12 aP$$ can also (for this polygon) be written $$P=\frac12 aP$$ (substituting $P$ instead of $A$). Assuming the polygon is not degenerate, you know that $P \neq 0$, so you can divide by $P$ on both sides of the equation. It is then easy to find the value of $a$.

(If degenerate polygons are allowed, then you get two possible values for $a$. It seems unlikely that this was the intended interpretation of the problem.)

A regular polygon, remember, is a polygon whose sides and interior angles are all congruent. To understand the formula for the area of such a polygon, some new vocabulary is necessary.

The center of a regular polygon is the point from which all the vertices are equidistant. The radius of a regular polygon is a segment with one endpoint at the center and the other endpoint at one of the vertices. Thus, there are n radii in an n-sided regular polygon. The center and radius of a regular polygon are the same as the center and radius of a circle circumscribed about that regular polygon.

An apothem of a regular polygon is a segment with one endpoint at the center and the other endpoint at the midpoint of one of the sides. The apothem of a regular polygon is the perpendicular bisector of whichever side on which it has its endpoint. A central angle of a regular polygon is an angle whose vertex is the center and whose rays, or sides, contain the endpoints of a side of the regular polygon. Thus, an n-sided regular polygon has n apothems and n central angles, each of whose measure is 360/n degrees. Every apothem is the angle bisector of the central angle that contains the side to which the apothem extends. Below are pictured these characteristics of a regular polygon.

Figure %: A regular polygon with a center (C), radius (r), apothem (a), and central angle

Once you have mastered these new definitions, the formula for the area of a regular polygon is an easy one. The area of a regular polygon is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter.

When an n-sided polygon is split up into n triangles, its area is equal to the sum of the areas of the triangles. Can you see how 1/2(ap) is equal to the sum of the areas of the triangles that make up a regular polygon? The apothem is equal to the altitude, and the perimeter is equal to the sum of the bases. So 1/2(ap) is only a slightly simpler way to express the sum of the areas of the n triangles that make up an n-sided regular polygon.

Figure %: Two n-sided polygons divided into n triangles

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Tim and Sam were finding the area of various regular polygons. Tim divided the polygons into triangles and was trying to calculate the area of the polygons, which took him too long. On the other hand, Sam found the area of the polygons easily, as he knew the length of the apothem. So let's learn about apothem today!

Before we get started, check out this interesting simulation to identify the apothem for various polygons. This apothem calculator will help you understand the lesson better.

Apothem is a line drawn from the center of any polygon to the midpoint of one of the sides.

The apothem formula, when the side length is given is:

$a$ = $\begin{align}\frac{S}{2\,\, \text{tan}\left ( \frac{180}{n} \right )}\end{align}$

Where,

$a$ = apothem length

$s$ = side length

$n$ = number of sides of a polygon.

The apothem formula , when the radius is given is:

$a$ = $r.cos\frac{180}{n}$

Where

$r$ = radius.

$n$ = number of sides

$Cos$ = cosine function which is calculated in degrees.

We can use the apothem area formula of a polygon to calculate the length of the apothem.

$A$ = area of the polygon

To calculate the area of a polygon with the help of apothem, we use the formula:

$A$ = $\dfrac{1}{2}aP$

Example: Find the area of a regular hexagon, if the side length is $5$ inches, and the apothem is $3$ inches.

$A$ = $\dfrac{1}{2}aP$

\[\begin{align}P &= [\text{side length}]\times [\text{no. of sides}]\\&= 5\times6 = 30\end{align}\]

After the perimeter is calculated, we use it in the formula of Area = $A$ = $\dfrac{1}{2}aP$

\[\begin{align}A &= \dfrac{1}{2}aP \\
& = \dfrac{1}{2} \left ( 3\right )\left (30 \right )\\& = 45 \text{ inches}^2\end{align}\]

The apothem is always perpendicular to the side on which it ends.
A regular polygon has all its sides and angles equal.

Solved Examples

Help Bryan find the length of the apothem of a regular pentagon of side = $10$ inches and area $150\sqrt{3}\,\,\text {inches}^{2}$.

Solution

Given,

$L$ = $10$ inches.

$A$ = $150\sqrt{3}\,\,\text {inches}^{2}$

So, the perimeter will be $P$ = $10\times 5$ = $50$ inches.

\[\begin{align}A&=\dfrac{1}{2}aP\\ 150\sqrt{3}&= \dfrac{1}{2}a\times 50\\ 25a&=150\sqrt{3}\\ a&=\dfrac{150\sqrt{3}}{25}\\

a&=6\sqrt{3}\text{ inches}\end{align}\]

Therefore, Apothem = $6\sqrt{3}$ inches.

$\therefore$ $a$ = $6\sqrt{3}$ inches

Can you help Dylan calculate the area of a regular hexagon of side 8 inches and apothem $4\sqrt{3}$, using the area of polygon formula?

Solution

Given,

side length = 8 inches.

perimeter $P$ = $L\times n$

= $8\times 6$

= $48$ inches

Area of polygon = \[\begin{align}A& = \dfrac{1}{2}aP\\&= \dfrac{1}{2}\times4\sqrt{3}\times 48\\&= 96\sqrt{3} \text{ inches}^2\end{align}\]

$\therefore$ The area of the polygon is $96\sqrt{3} \text{ inches}^2$

Emily's teacher asked her to calculate the area of a regular hexagon, whose apothem is 7 inches and perimeter 48 inches.

Solution

Given,

$a$ = 7 inches.

$P$ = 48 inches.

Thus, the Area of hexagon will be:

\[\begin{align}&\frac{1}{2}aP\\ &=\frac{1}{2}\times 7\times 48\\ &=\frac{1}{2}\times 336\\ &=\frac{336}{2}\\

&=168 \text{ inches}^2\end{align}\]

$\therefore$ $A$ = $168 \text{ inches}^2$

Can you help Jose calculate the length of the apothem of a square, which has a side length of 3 inches?

Solution

The formula for calculating apothem, when side length is given is:

$a$ = $\begin{align}\frac{S}{2 \,\,\text{tan}\left ( \frac{180}{n} \right )}\end{align}$

\[\begin {align}a &=\dfrac{S}{2\,\,\text{tan}\left ( \dfrac{180}{n} \right)}\\ &=\dfrac{3}{2\,\,\text{tan}\left ( \dfrac{180}{4} \right)}\\ &=\dfrac{3}{2\,\,\text{tan} 45}\\ &=\dfrac{3}{2\times 1}\\ &=\dfrac{3}{2}\\

&=1.5\text{ inches}\end{align}\]

$\therefore$ The length of apothem is $1.5\text{ inches}$

Devin wanted to calculate the area of a regular octagon of side 12 inches and apothem 14.5 inches. Help him find the answer.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

We hope you enjoyed learning about apothem with the simulations and practice questions. Now you will be able to easily solve problems related to the apothem.

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