# The Circumference of a Circle

🏆Practice circumference

The circumference is actually the length of the circular line. It is calculated by multiplying the radius by 2, which has an approximate value of π. It can also be said that the circumference is equal to the the diameter of the circumference multiplied by π (since the diameter is actually twice the radius of the circumference). It is customary to identify the circumference (the perimeter) with the letter P.

### $P=2\times\pi\times R$

We will illustrate the concept with a simple example. Here is a circle, as shown in the drawing in front of you:

The radius of the circumference is $3\text{ cm}$.

You can calculate the circumference of the circle by placing the data:

$P=2\times R\timesπ=2\times3\times3.14=18.84$

That is, the circumference is $18.84\text{ cm}$.

## Test yourself on circumference!

$$r=2$$

Calculate the circumference.

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## Circumference Exercises

### Exercise 1

Given the circle whose radius $3\operatorname{cm}$

What is its circumference?

Solution

We use the formula of the circumference $2\pi r$

Replace the given radius accordingly

$p=2\pi\cdot3=6\pi$

$6\pi$

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### Exercise 2

Given the circle in the figure

What is its diameter?

Solution

We use the formula of the circumference $2\pi r$

Replace the data accordingly

$16\pi=2\pi\cdot r$

Divide by $2\pi$

$\frac{16\pi}{2\pi}=r$

Reduce by $\pi$

$r=8$

Diameter= Radius multiplied by $2$

$2\cdot8=16$

$16$

### Exercise 3

Given the circle in the figure

Is it possible to find the circumference?

Solution

The given chord of the circle is not the diameter or the radius and no other data is given other than the chord in the figure.

It is not possible to calculate a circle without some data about the radius or the diameter or without other information that helps to find them.

It is not possible to find the circumference

Do you know what the answer is?

### Exercise 4

Question

What is the radius of the circle whose circumference is $9a\pi$ cm?

Solution

We use the formula of the circumference $2\pi r$

Replace accordingly

$2\pi r$

Divide by $2\pi$

$\frac{9a\pi}{2\pi}=r$

Reduce by pi

$r=4.5a$

$4.5a$

### Exercise 5

Given the shape in the figure

The quadrilateral is a square in which each side is extended by a quarter circle, the quarters of the circle being identical.

Given that the total circumference of the shape is $24+12\pi$ cm.

What are the lengths of the sides of the square?

Solution

The parts marked by R are the radii of the 4 circles.

We calculate the circumference of the shape

The marked sides are part of the circumference

$\left(forma\right)P=4\cdot\frac{1}{4}P\left(círculo\right)+4r\left(círculo\right)$

$\left(forma\right)P=P\left(forma\right)+4R\left(círculo\right)$

$\left(forma\right)P=2\pi R+4R$

$24+12\pi=2\pi R+4R$

$24+12\pi=R(2\pi+4)$

Divide by $2\pi+4$

$\frac{24+12\pi}{2\pi+4}=R$

$\frac{6(2\pi+4)}{2\pi+4}=R$

Divide by $2\pi+4$

$R=6$

$6$

### Exercise 6

Given the circle in the figure:

The radius is equal to $4\text{ cm}$

What is its circumference?

Solution

Since we know the radius, all we have to do is replace the data in the formula to calculate the circumference of the circle:

$P=2\times\pi\times R$

$P=2\times3.14\times4=25.12$

$8\pi$ o $25.12\text{ cm}$

### Exercise 7

Given the circle whose radius has a length of $9$ cm

What is its circumference?

Solution

Since we know the radius, all we have to do is replace the data in the formula to calculate the circumference of the circle:

$P=2\times\pi\times R$

$P=2\times3.14\times9=56.52$

$56.52\text{ cm}$

Do you think you will be able to solve it?

### Exercise 8

Given the circle whose diameter is $12\text{ cm}$

What is its circumference?

Solution

We know the diameter of the circle, to calculate its circumference we must find the radius.

The diameter of the circle is twice the radius, so we can conclude that half of the diameter is the radius:

$12:2=R=6$

We put the result in the formula to calculate the circumference of the circle and we will get the answer:

$P=2\times\pi\times R$

$P=2\times3.14\times6=37.68$

$12\pi$ or $37.68$ cm

### Exercise 9

A bicycle has tires with a radius of $40$ cm,

the wheels made five complete turns.

How far did the bicycle travel?

Solution

First we calculate the circumference of the wheels of the bicycle.

We know that the radius is $40$ cm, so we will put the radius in the formula to calculate the circumference.

$P=2\times\pi\times R$

$P=2\times3.14\times40$

$P=2\times3.14\times40=251.2$

Now that we know that the circumference of the wheels is $251.2$ cm, we can calculate the distance they traveled by multiplying the circumference by the number of turns:

$5\times251.2=1256$

Since we want to know the distance in meters we will divide it by $100$

$\frac{1256}{100}=12.56$

$12.56$ meters

### Exercise 10

For a scientific experiment, Sebastian needs to produce a wheel that turns exactly $17$ times around a track $6.8$ m long.

What should the radius of the wheel be?

Solution

To solve, let's first understand the question.

For the wheel to make $17$ turns in a distance of $6.8$ mts, the circumference must be equal to:

$\frac{680}{17}=40$

That is, the circumference is equal to $40$

The question is what is the radius of the circle and therefore we put the data we have in the formula for calculating the circumference.

$P=2\times\pi\times R$

$40=2\times3.14\times R$

$2\times3.14=6.28$

$40=6.28R$

$\frac{40}{6.28}=R$

$R=6.36$

$R=6.36$ meters.

## Review questions

What is the circumference?

As we know, the perimeter of a figure length of all of the sides of that figure, in the case of the circle its perimeter, called the circumference, is the measure or length of the entire circular line.

How to measure the circumference of a circle?

To calculate the circumference we have two formulas that we can use:

$P=2\pi r$

Where

$P$ is the perimeter

$\pi=3.14$

$r$ is the radius

By definition we know that the radius is half the diameter or the diameter is twice the radius, then according to what $D=2r$, we can use the following formula

$P=\pi D$

What is an example of finding the circumference?

Example

Calculate the circumference of the circle, given that $r=5\text{ cm}$

Solution:

To calculate the circumference, we will use that the radius $r=5\text{ cm}$ and just substitute in our formula:

$P=2\pi r$

$P=2\times3.14\times5\operatorname{cm}$

$P=6.28\times5\operatorname{cm}$

$P=31.4\operatorname{cm}$

Ó

$P=2\times\pi\times5\operatorname{cm}$

$P=10\pi\text{ cm}$

Solution

$P=10\pi\text{ cm}$

Ó

$P=31.4\operatorname{cm}$

Do you know what the answer is?

## Examples with solutions for Circumference

### Exercise #1

Look at the circle in the figure:

Its radius is equal to 4.

What is its circumference?

### Step-by-Step Solution

The formula for the circumference is equal to:

$2\pi r$

### Exercise #2

Look at the circle in the figure.

What is its circumference if its radius is equal to 6?

### Step-by-Step Solution

Formula of the circumference:

$P=2\pi r$

We insert the given data into the formula:

$P=2\times6\times\pi$

$P=12\pi$

$12\pi$

### Exercise #3

O is the center of the circle in the figure below.

What is its circumference?

### Step-by-Step Solution

We use the formula:$P=2\pi r$

We replace the data in the formula:$P=2\times8\pi$

$P=16\pi$

$16\pi$ cm

### Exercise #4

Is it possible that the circumference of a circle is 8 meters and its diameter is 4 meters?

### Step-by-Step Solution

To calculate, we will use the formula:

$\frac{P}{2r}=\pi$

Pi is the ratio between the circumference of the circle and the diameter of the circle.

The diameter is equal to 2 radii.

Let's substitute the given data into the formula:

$\frac{8}{4}=\pi$

$2\ne\pi$

Therefore, this situation is not possible.

Impossible

### Exercise #5

A circle has a circumference of 31.41.

What is its radius?

### Step-by-Step Solution

To solve the exercise, first we must remember the circumference formula:

$P= 2\pi R$

P is the circumference and Pi has a value of 3.14 (approximately).

We substitute in the known data:

$31.41=2\cdot3.141\cdot R$

Keep in mind that the result can be easily simplified using Pi:

$\frac{31.41}{3.141}=2R$

$10=2R$

Finally, we simplify by 2:

$5=R$