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 $3+3+3+3 = 12.$

Any closed figure has a perimeter, such as the irregular polygon to the right with perimeter $1+1+2+ 3= 7.$ Some shapes do not have a finite number of sides, so calculating the perimeter can be less straighforward. For example, the perimeter (or circumference) of a circle is $2\pi r,$ where $r$ is the radius of the circle. Fascinatingly, some shapes (such as certain fractals) have infinite perimeter, despite having a finite area.

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

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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?

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

$$6 + 6 + 6 + 6 + 6= 5 \cdot 6 = 30 . \ _\square$$

What is the perimeter of a rectangle with width $4$ and height $7?$

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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 $\angle B= 90^{\circ}.$

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First, use the Pythagorean theorem to find the length of $AC:$

$$AC^{2}=AB^{2}+BC^{2}=3^{2}+4^{2} \implies AC=5.$$

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?

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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.

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We have

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

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 $14\text{ cm}?$

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We have

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

Find the perimeter of a semicircle of radius $21$ cm.

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We have

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

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.

In this section, we will learn about applications of the concept of perimeter in real life. They are mostly of fencing-type problems. Here is an example:

Mr. Antony has a rectangular garden of area $630\text{ m}^2$ and length $42\text{ m}.$ To save his garden from domestic animals grazing them, he wants to fence his whole garden. Find much Mr. Antony has to spend for fencing his field at the rate of $3 per meter?

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Given, the area of the rectangle $A = 630\text{ m}^2$ and its length $l = 42\text{ m},$ we know that

$$A = l \times b \implies b = \dfrac{A}{l} = \dfrac{630}{42} = 15\ (\text{m}).$$

Now, let's find perimeter: $P = 2(l + b) = 2(42 + 15) = 114\ (\text{m}).$

Since the cost of fencing per meter is $3, the total cost of fencing the whole feild is $3 \times P = 3 \times 114 = 342$ dollars. $_\square$