# Perimeter

🏆Practice circumference

## What is the perimeter?

The perimeter indicates the distance we will walk if we start from a certain point, complete a full lap, and return exactly to the starting point.
For example, if we are asked what the perimeter of the waist is, we will take a tape measure and measure the perimeter from a certain point until completing a full lap and returning to the same point from which we started the measurement.
It works exactly the same way in mathematics. The perimeter of any shape is the distance from a specific point back to it after having completely surrounded it.
If this is our figure:

Its perimeter will be the distance we cover if we travel along its line from a certain point, and return to it after making a full lap. Imagine that you are surrounding the figure:

## Test yourself on circumference!

A circle has a radius of 3 cm.

What is its perimeter?

## Units of Measurement for Perimeter

The perimeter is measured in units of mm, cm, or meters, according to what the question states.
Generally, most figures are given in units of cm.
We can convert the different units of measure in the following way:
$1$ cm = $10$ mm
$1$ meter = $100$ cm

Now we will learn to calculate the perimeter of the most known figures. Are we ready?
How is the perimeter calculated in general?
All the lengths of the edges (or sides) of the figure are added together.
The sum of all the edges is the perimeter.

## Perimeter of the Square

$a$ -> Side of the square
In the square, all sides are equal, therefore, its perimeter will be $4$ times the side $a$.
We will multiply the side of the square by $4$.

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## Perimeter of the Rectangle

Let's add up the sides of the rectangle. The opposite sides are equal.

## Perimeter of the triangle

Let's add up all the sides of the triangle.
In an isosceles triangle it is enough to know the length of the base and one of the two equal sides.
In an equilateral triangle it is enough to know the length of one side.

Do you know what the answer is?

## Perimeter of a Rhombus

$a$ -> side of the rhombus
In the rhombus all sides are equal, therefore, its perimeter will be 4 times the side a.

## Perimeter of a Parallelogram

Let's add up the sides of the parallelogram. The opposite sides are equal.

## Circle Perimeter

$r$ –>  Radius of the circumference
Pi $π$ will be calculated as the number –> $3.14$

Let's multiply Pi –> $$3.14$$by the radius by $2$

## Perimeter of the Trapezoid

Let's add up all the sides of the trapezoid

Do you think you will be able to solve it?

## Perimeter of the deltoid

In the deltoid, the 2 adjacent sides are equal.

## Perimeter of Composite Figures

The key to calculating the perimeter of these figures is to add up absolutely all the sides without forgetting any of them.
Start on one side, follow the entire round and stop when you reach the same side from which you started.

Let's see an example:

The area is:
$4+4+4+5+2+8+2+3=32$

## What is the difference between perimeter and surface area?

The perimeter is measured in two-dimensional figures that do not have volume, for example, a rectangle

In contrast, the surface area is measured in three-dimensional figures that do have volume, for example, a cylinder or cube.

## Examples and exercises with solutions for the perimeter of the parallelogram

### examples.example_title

Given a parallelogram in which the length of one side is 4 times greater than the length of another side and given that the length of the longer side is X:

Express the perimeter of the parallelogram in terms of X.

### examples.explanation_title

In a parallelogram, each pair of opposite sides are equal and parallel: AB=CD and AC=BD

Given that the length of one side is 4 times greater than the other side equal to X, we can affirm that:

$AB=CD=4AC=4BD$

Now we replace the data in this equation (assuming that AB=CD=X):

$x=x=4AC=4BD$

We divide by 4:

$\frac{x}{4}=\frac{x}{4}=AC=BD$

Now we calculate the perimeter of the parallelogram and express both AC and BD using X:

$P=x+\frac{x}{4}+x+\frac{x}{4}$

$P=2x+\frac{x}{4}+\frac{x}{4}=2\frac{1}{2}x$

2.5X

### examples.example_title

ABCD is a parallelogram whose perimeter is equal to 38 cm.

AB is greater than CE by 2

AD is less than CE by 3

CE height of the parallelogram for the side AD

Calculate the area of the parallelogram

### examples.explanation_title

Let's call CE as X

According to the data

$AB=x+2,AD=x-3$

The perimeter of the parallelogram:

$2(AB+AD)$

$38=2(x+2+x-3)$

$38=2(2x-1)$

$38=4x-2$

$38+2=4x$

$40=4x$

$x=10$

Now it can be argued:

$AD=10-3=7,CE=10$

The area of the parallelogram:

$CE\times AD=10\times7=70$

70 cm²

## Examples and exercises with solutions for the perimeter of a trapezoid

### examples.example_title

Given the isosceles triangle ABC,

The side AD is the height in the triangle ABC

and inside it, EF is drawn:

AF=5 AB=17

What is the perimeter of the trapezoid EFBC?

### examples.explanation_title

To find the perimeter of the trapezoid, all its sides must be added:

We will focus on finding the bases.

To find GF we use the Pythagorean theorem: $A^2+B^2=C^2$in the triangle AFG

We replace

$3^2+GF^2=5^2$

We isolate GF and solve:

$9+GF^2=25$

$GF^2=25-9=16$

$GF=4$

We perform the same process with the side DB of the triangle ABD:

$8^2+DB^2=17^2$

$64+DB^2=289$

$DB^2=289-64=225$

$DB=15$

We start by finding FB:

$FB=AB-AF=17-5=12$

Now we reveal EF and CB:

$GF=GE=4$

$DB=DC=15$

This is because in an isosceles triangle, the height divides the base into two equal parts so:

$EF=GF\times2=4\times2=8$

$CB=DB\times2=15\times2=30$

All that's left is to calculate:

$30+8+12\times2=30+8+24=62$

62

## Examples and exercises with solutions for the perimeter of the triangle

### examples.example_title

Given an equilateral triangle:

The perimeter of the triangle is 33 cm, what is the value of X?

### examples.explanation_title

We know that in an equilateral triangle all sides are equal,

Therefore, if we know that one side is equal to X, all sides are equal to X.

We know that the perimeter of the triangle is 33.

The perimeter of the triangle is equal to the sum of the sides together.

We replace the data:

$x+x+x=33$

$3x=33$

We divide the two sections by 3:

$\frac{3x}{3}=\frac{33}{3}$

$x=11$

11

### examples.example_title

Given the triangle in the figure

Given that the perimeter is $12+4\sqrt{5}$ cm

What is the length of the hypotenuse?

### examples.explanation_title

We calculate the perimeter of the triangle:

$12+4\sqrt{5}=4+AC+BC$

As we want to find the hypotenuse, that is BC, we isolate it:

$12+4\sqrt{5}-4-AC=BC$

$BC=8+4\sqrt{5}-AC$

Find AC using the Pythagorean theorem:

$AB^2+AC^2=BC^2$

$4^2+AC^2=(8+4\sqrt{5}-AC)^2$

$16+AC^2=(8+4\sqrt{5})^2-2\times AC(8+4\sqrt{5})+AC^2$

We will reduce the two$AC^2$

$16=8^2+2\times8\times4\sqrt{5}+(4\sqrt{5})^2-2\times8\times AC-2AC4\sqrt{5}$

$16=64+64\sqrt{5}+16\times5-16AC-8\sqrt{5}AC$

$16AC+8\sqrt{5}AC=64+64\sqrt{5}+16\times5-16$

$AC(16+8\sqrt{5})=128+64\sqrt{5}$

$AC=\frac{128+64\sqrt{5}}{16+8\sqrt{5}}=\frac{8(16+8\sqrt{5})}{16+8\sqrt{5}}$

We reduce and obtain

$AC=8$

Now we can replace AC with the value we found for BC:

$BC=8+4\sqrt{5}-AC$

$BC=8+4\sqrt{5}-8=4\sqrt{5}$

### examples.solution_title

$4\sqrt{5}$ cm

## Examples and exercises with solutions for the perimeter of the rectangle

### examples.example_title

Given the following rectangle:

Find the perimeter.

### examples.explanation_title

Since in a rectangle all pairs of opposite sides are equal:

$AD=BC=5$

$AB=CD=9$

Now we calculate the perimeter of the rectangle by adding the sides:

$5+5+9+9=10+18=28$

28

### examples.example_title

Given the rectangle composed of two squares:

What is its perimeter?

### examples.explanation_title

In a square, all sides are equal. Therefore:
$AB+BC+CD+DE+EF+FA=6$

Thus, we find out what the side AC is equal to:

$AC=AB+BC$

$AB=6+6=12$

In a rectangle, we know that the opposite sides are equal to each other, therefore:

$AB=FD=12$

Therefore, the formula for the perimeter of the rectangle will look like this:

$2\times AB+2\times CD$

We replace the data:

$2\times12+2\times6=$

$24+12=36$

### examples.solution_title

36

Do you know what the answer is?
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