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.

The formula for calculating the circumference is:

P=2×π×R P=2\times\pi\times R

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Test yourself on circumference!

einstein

\( r=2 \)

Calculate the circumference.

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

perimeter of the circumference

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

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

P=2×R×π=2×3×3.14=18.84 P=2\times R\timesπ=2\times3\times3.14=18.84

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


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Circumference Exercises

Exercise 1

Task

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

What is its circumference?

Exercise 1 - Given the circle whose radius is 3 cm

Solution

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

Replace the given radius accordingly

p=2π3=6π p=2\pi\cdot3=6\pi

Answer

6π 6\pi


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Exercise 2

Task

Given the circle in the figure

What is its diameter?

Exercise 2- Given the circle in the figure

Solution

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

Replace the data accordingly

16π=2πr 16\pi=2\pi\cdot r

Divide by 2π 2\pi

16π2π=r \frac{16\pi}{2\pi}=r

Reduce by π \pi

r=8 r=8

Diameter= Radius multiplied by 2 2

28=16 2\cdot8=16

Answer

16 16


Exercise 3

Task

Given the circle in the figure

Is it possible to find the circumference?

exercise 3 - Given the circle in the figure

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.

Answer

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 9aπ 9a\pi cm?

Solution

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

Replace accordingly

2πr 2\pi r

Divide by 2π 2\pi

9aπ2π=r \frac{9a\pi}{2\pi}=r

Reduce by pi

r=4.5a r=4.5a

Answer

4.5a 4.5a


Exercise 5

Task
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+12π 24+12\pi cm.

What are the lengths of the sides of the square?

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

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

(forma)P=414P(cıˊrculo)+4r(cıˊrculo)\left(forma\right)P=4\cdot\frac{1}{4}P\left(círculo\right)+4r\left(círculo\right)

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

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

24+12π=2πR+4R 24+12\pi=2\pi R+4R

24+12π=R(2π+4) 24+12\pi=R(2\pi+4)

Divide by 2π+4 2\pi+4

24+12π2π+4=R \frac{24+12\pi}{2\pi+4}=R

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

Divide by 2π+4 2\pi+4

R=6 R=6

Answer

6 6


Check your understanding

Exercise 6

Task

Given the circle in the figure:

Exercise 6 Given the circle in the figure

The radius is equal to 4 cm 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×π×R P=2\times\pi\times R

P=2×3.14×4=25.12 P=2\times3.14\times4=25.12

Answer

8π 8\pi o 25.12 cm 25.12\text{ cm}


Exercise 7

Task

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

What is its circumference?

Exercise 7 - Given the circle whose radius has a length of 9 cm

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×π×R P=2\times\pi\times R

P=2×3.14×9=56.52 P=2\times3.14\times9=56.52

Answer

56.52 cm 56.52\text{ cm}


Do you think you will be able to solve it?

Exercise 8

Task

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

What is its circumference?

Exercise 8 - Given the circle whose diameter is 12

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 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×π×R P=2\times\pi\times R

P=2×3.14×6=37.68 P=2\times3.14\times6=37.68

Answer

12π 12\pi or 37.68 37.68 cm


Exercise 9

Task

A bicycle has tires with a radius of 40 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 40 cm, so we will put the radius in the formula to calculate the circumference.

P=2×π×R P=2\times\pi\times R

P=2×3.14×40 P=2\times3.14\times40

P=2×3.14×40=251.2 P=2\times3.14\times40=251.2

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

5×251.2=1256 5\times251.2=1256

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

1256100=12.56 \frac{1256}{100}=12.56

Answer

12.56 12.56 meters


Test your knowledge

Exercise 10

Task

For a scientific experiment, Sebastian needs to produce a wheel that turns exactly 17 17 times around a track 6.8 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 17 turns in a distance of 6.8 6.8 mts, the circumference must be equal to:

68017=40 \frac{680}{17}=40

That is, the circumference is equal to 40 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×π×R P=2\times\pi\times R

40=2×3.14×R 40=2\times3.14\times R

2×3.14=6.28 2\times3.14=6.28

40=6.28R 40=6.28R

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

R=6.36 R=6.36

Answer:

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=2πr P=2\pi r

Where

P P is the perimeter

π=3.14 \pi=3.14

r 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 D=2r , we can use the following formula

P=πD P=\pi D


What is an example of finding the circumference?

Example

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

Calculate the perimeter of the circumference with r=5

Solution:

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

P=2πr P=2\pi r

P=2×3.14×5cm P=2\times3.14\times5\operatorname{cm}

P=6.28×5cm P=6.28\times5\operatorname{cm}

P=31.4cm P=31.4\operatorname{cm}

Ó

P=2×π×5cm P=2\times\pi\times5\operatorname{cm}

P=10π cm P=10\pi\text{ cm}

Solution

P=10π cm P=10\pi\text{ cm}

Ó

P=31.4cm P=31.4\operatorname{cm}


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