# Perimeter

The **perimeter** of a two-dimensional figure is the length of the boundary of the figure. If the figure is a polygon such as a triangle, square, rectangle, pentagon, etc., then the perimeter is the sum of the edge lengths of the polygon. For example, a square with side length 3 has perimeter \(3+3+3+3 = 12.\)

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## Perimeter of Regular Polygons

## What is the perimeter of a square with side length 6?

Since a square has four sides of equal length and the perimeter of a square is the sum of its side lengths, the perimeter of the square is

\[ 6 + 6 + 6 + 6 = 24. \ _\square \]

You can also do \(6 \times 4\) on the count that it is a square.

## What is the perimeter of a regular pentagon whose side lengths are all \(6\)?

Since a regular pentagon has five sides and all its side lengths are \(6\), the perimeter of the pentagon is

\[ 6 + 6 + 6 + 6 + 6= 5 \cdot 6 = 30 . \ _\square \]

## Perimeter of Irregular Polygons

## What is the perimeter of a rectangle with width \(4\) and height \(7\)?

Since a rectangle has two sides with equal width and two sides with equal height, and the perimeter of a rectangle is the sum of its side lengths, the perimeter of the rectangle is

\[ 4 + 4 + 7 + 7 = 2(4) + 2(7) = 8 + 14 = 22 . \ _\square \]

## Find the perimeter of a triangle \(ABC\) with \(AB= 3\text{ cm}, BC=4\text{ cm},\) and \(\angle B= 90^{\circ}.\)

First, use Pythagorean theorem to find the length of \(AC:\)

\[AC^{2}=AB^{2}+BC^{2}=3^{2}+4^{2} \Rightarrow AC=5.\]

Then add \(AB,BC,CA\) to get the perimeter, which is

\[3+4+5 =12\text{ (cm)}.\ _\square\]

## Suppose an ant walks along the boundary of a triangle with side lengths \(4, 6,\) and \(7\). If the ant starts at one vertex of the triangle and makes \(3\) complete trips around the triangle boundary to return to its starting point, what is the total distance traveled by the ant?

Since the perimeter is the sum of edge lengths, the perimeter of the triangle is \(4 + 6 + 7 = 17\). Since the ant makes \(3\) complete trips around the triangle boundary, the total distance traveled by the ant is

\[ 17 \times 3 = 51. \ _\square \]

Find the perimeter of the given figure.

We have

\[\begin{align}

\text{Perimeter} & = \text{Sum of side lengths}\\ & = 9 + 8 + 7 + 3 + 4 + 5\\ & = 36.\ _\square \end{align}\]

## Perimeter of Other Shapes (Circles, Fractals, etc.)

The perimeter of a circle with radius \(r\) is \(2\pi r\). The perimeter of an arc of a circle subtending angle \(\theta\) (in degrees) at the center is \(2\pi r \times \frac {\theta}{360}\).

What is the perimeter (circumference) of a circle of diameter \(14\text{ cm}?\)

We have

\[\begin{align}

\text{Radius} & = \dfrac{\text{Diameter}}{2}\\ & = \dfrac{14}{2}\\ & = 7\text{ (cm)}\\ \\ \text{Circumference} & = 2× \pi × r\\ & = 2 × \dfrac{22}{7} × 7\\ & = 44\text{ (cm)}. \end{align}\]

Find the perimeter of a semicircle of radius \(21\)cm.

We have

\[\begin{align}

\text{Perimeter of a semicircle} & = \left( \dfrac{1}{2} × 2 × \pi × r \right) + 2r\\ & = \left( \dfrac{1}{2} × 2× \dfrac{22}{7} × 21\right) + 2×21\\ & = 66 + 42\\ & = 106\text{ (cm)}. \end{align}\]

In the figure to the right, a circle is circumscribed around a regular hexagon, and the same circle is inscribed within another regular hexagon.

Let \(P_1\) be the perimeter of the larger hexagon, let \(P_2\) be the perimeter of the smaller hexagon, and let \(C\) be the circumference of the circle.

The circumference of the circle can be approximated by finding the mean of the two perimeters: \[C\approx \frac{P_1+P_2}{2}.\] If \(\pi\) is approximated using the approximation for circumference above, then \(\displaystyle\pi\approx\frac{a}{b}+\sqrt{c}\), where \(a, b, c\) are positive integers, \(a\) and \(b\) are coprime, and \(c\) is square-free.

Find \(a+b+c\).

## Perimeter using Polar Coordinates

We can calculate the perimeter of an enclosed figure if its graph on a Cartesian plane is a function in polar coordinates. Specifically, suppose the figure has the equation \(r = f(\theta)\) such that \(f\) is a continuous, nonnegative function for \(\theta \in [0, 2\pi]\) and \(f(0) = f(2\pi)\). Then its perimeter is given by the following formula:

\[P = \int_{0}^{2\pi} \sqrt{r^2 + \left(\dfrac{dr}{d\theta}\right)^2} d\theta.\]

Suppose we want to calculate the circumference of a circle with radius \(r\). Then \(r\) is a constant, and its derivative is \(0\). So the formula gives us

\[P = \int_{0}^{2\pi} \sqrt{r^2 + 0^2} d\theta = \int_{0}^{2\pi} r d\theta = r(2\pi-0) = 2\pi r.\]

Therefore the circumference is \(2\pi r.\)

Unfortunately, for other shapes, this formula usually leads to complicated integrals which often do not have a simple closed form.

## Problem Solving

A red rectangle is reshaped into a blue square, as shown below, such that both quadrilaterals have the same perimeter. The area of the square is \(4\text{ cm}^2\) larger than that of the rectangle.

If all the sides of both quadrilaterals have prime number lengths \((\)in \(\text{cm}),\) then what is the perimeter of the square?