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

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

C - 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} .


Practice Circumference

Examples with solutions for Circumference

Exercise #1

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

888OOO What is its circumference?

Video Solution

Step-by-Step Solution

We use the formula:P=2πr P=2\pi r

We replace the data in the formula:P=2×8π P=2\times8\pi

P=16π P=16\pi

Answer

16π 16\pi cm

Exercise #2

Look at the circle in the figure:

444

Its radius is equal to 4.

What is its circumference?

Video Solution

Step-by-Step Solution

The formula for the circumference is equal to:

2πr 2\pi r

Answer

Exercise #3

Look at the circle in the figure.

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

6

Video Solution

Step-by-Step Solution

Formula of the circumference:

P=2πr P=2\pi r

We insert the given data into the formula:

P=2×6×π P=2\times6\times\pi

P=12π P=12\pi

Answer

12π 12\pi

Exercise #4

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

Video Solution

Step-by-Step Solution

To calculate, we will use the formula:

P2r=π \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:

84=π \frac{8}{4}=\pi

2π 2\ne\pi

Therefore, this situation is not possible.

Answer

Impossible

Exercise #5

Look at the circle in the figure.

The radius of the circle is 23 \frac{2}{3} .

What is its perimeter?

Video Solution

Step-by-Step Solution

The radius is a straight line that extends from the center of the circle to its outer edge.

The radius is essential for calculating the circumference of the circle, according to the following formula:

If we substitute the radius we currently have, the formula will be:

2*π*2/3

Let's start solving, we'll rearrange the formula:

π*2*2/3 =

We'll multiply the fraction by the whole number:

π*(2*2)/3 =

π*4/3 =

4/3π

And that's the result!

Answer

43π \frac{4}{3}\pi

Exercise #6

The circumference of a circle is 14.

How long is the circle's radius?

Video Solution

Step-by-Step Solution

We begin by using the formula:

P=2πr P=2\pi r

We then insert the given data into the formula:

14=2×π×r 14=2\times\pi\times r

Lastly we divide Pi by 2:

142π=2πr2π \frac{14}{2\pi}=\frac{2\pi r}{2\pi}

7π=r \frac{7}{\pi}=r

Answer

7π \frac{7}{\pi}

Exercise #7

A circle has a circumference of 31.41.

What is its radius?

Video Solution

Step-by-Step Solution

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

P=2πR 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=23.141R 31.41=2\cdot3.141\cdot R

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

31.413.141=2R \frac{31.41}{3.141}=2R

10=2R 10=2R

Finally, we simplify by 2:

5=R 5=R

Answer

5

Exercise #8

A circle has a circumference of 50.25.

What is its radius?

Video Solution

Step-by-Step Solution

We use the formula:

P=2πr P=2\pi r

We insert the known data into the formula:

50.25=3.14×2r 50.25=3.14\times2r

50.25=2×r×3.14 50.25=2\times r\times3.14

50.25=6.28r 50.25=6.28r

50.256.28=6.28r6.28 \frac{50.25}{6.28}=\frac{6.28r}{6.28}

r=8 r=8

Answer

8

Exercise #9

The area of the rectangle in the drawing is 28X cm².

What is the area of the circle?

S=28XS=28XS=28X777

Video Solution

Step-by-Step Solution

Let's draw the center of the circle and we can divide the diameter of the circle into two equal radii

Now let's calculate the length of the radii as follows:

7×2r=28x 7\times2r=28x

14r=28x 14r=28x

We'll divide both sides by 14:

r=2814x r=\frac{28}{14}x

r=2x r=2x

Let's calculate the circumference of the circle:

P=2π×r=2π×2x=4πx P=2\pi\times r=2\pi\times2x=4\pi x

Answer

4πx 4\pi x

Exercise #10

Below is a circle bounded by a parallelogram:

36

All meeting points are tangential to the circle.
The circumference is 25.13.

What is the area of the parallelogram?

Video Solution

Step-by-Step Solution

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

AE=AF=3 AE=AF=3
BG=BF=6 BG=BF=6

And from here we can calculate:

AB=AF+FB=3+6=9 AB=AF+FB=3+6=9

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

Since the circumference is 25.13.

Circumference formula:2πR 2\pi R
We replace and solve:

2πR=25.13 2\pi R=25.13
πR=12.565 \pi R=12.565
R4 R\approx4

The height of the parallelogram is equal to two radii, that is, 8.

And from here you can calculate with a parallelogram area formula:

AlturaXLado AlturaXLado

9×872 9\times8\approx72

Answer

72 \approx72

Exercise #11

The following is a circle enclosed in a parallelogram:

36

All meeting points are tangent to the circle.
The circumference is 25.13.

What is the area of the zones marked in blue?

Video Solution

Step-by-Step Solution

First, we add letters as reference points:

Let's observe points A and B.

We know that two tangent lines to a circle that start from the same point are parallel to each other.

Therefore:

AE=AF=3 AE=AF=3
BG=BF=6 BG=BF=6

From here we can calculate:

AB=AF+FB=3+6=9 AB=AF+FB=3+6=9

Now we need the height of the parallelogram.

We know that F is tangent to the circle, so the diameter that comes out of point F will also be the height of the parallelogram.

It is also known that the diameter is equal to two radii.

It is known that the circumference of the circle is 25.13.

Formula of the circumference:2πR 2\pi R
We replace and solve:

2πR=25.13 2\pi R=25.13
πR=12.565 \pi R=12.565
R4 R\approx4

The height of the parallelogram is equal to two radii, that is, 8.

And from here it is possible to calculate the area of the parallelogram:

Lado x Altura \text{Lado }x\text{ Altura} 9×872 9\times8\approx72

Now, we calculate the area of the circle according to the formula:πR2 \pi R^2

π42=50.26 \pi4^2=50.26

Now, subtract the area of the circle from the surface of the trapezoid to get the answer:

7256.2421.73 72-56.24\approx21.73

Answer

21.73 \approx21.73

Exercise #12

r=11 r=11

Calculate the circumference.

111111

Video Solution

Answer

69.115

Exercise #13

r=7 r=7

Calculate the circumference.

777

Video Solution

Answer

43.982

Exercise #14

r=2 r=2

Calculate the circumference.

222

Video Solution

Answer

12.56

Exercise #15

r=6 r=6

Calculate the circumference.

666

Video Solution

Answer

37.699

Topics learned in later sections

  1. Circle
  2. Diameter
  3. Pi
  4. The Center of a Circle
  5. Radius
  6. How is the radius calculated using its circumference?
  7. Perimeter
  8. Area
  9. Elements of the circumference
  10. Area of a circle