# Elements of the circumference

🏆Practice circle

## What is circumference?

This question is not easy to answer and even more complicated to understand. If you imagine any point on a flat surface and a series of points whose distance from that point is identical, then you are looking at a circle.

## Test yourself on circle!

$$r=2$$

Calculate the circumference.

## Circumference characteristics

### Circumference perimeter

The perimeter of any circumference can be calculated. Generally, we can say that to calculate it, we must multiply by $2$ the value of $π$ (pi) and the length of the radius.

Click to access the article on the perimeter of the circumference.

## Circle area

Another equally important data that we can obtain with respect to any circumference is the area of the circle. To find it, we must raise the length of the radius squared and then multiply the result obtained by π. Click here to access the article on the area of the circle.

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## Examples and practice

### Exercise 1

Let's consider the following circle.

The radius of the circle is equal to $7 cm$.

Use the image and the data provided to calculate:

1. the diameter of the circumference

2. the perimeter of the circle

3. the area of the circle

Solution:

$P=2\times R\timesπ=2\times7\times3,14=43,96$

The perimeter of the circumference equals $43,96 cm$.

3. To calculate the area of the space inside the circumference, i.e., the circle, we must square the length of the radius of the circumference and then multiply the result obtained by the value of $π$.

Thus, we obtain:

$S=π\times R\times R=3,14X7\times7=153,86$

The area of the circle is $153,86cm²$

The diameter of the circle equals $14 cm$.

### Exercise 2

Let's consider the following circle.

We know that its diameter is $20cm$.

Use the image and the data provided to calculate:

1. the radius of the circle
2. the perimeter of the circle
3. the area of the circle

Solution:

1. The diameter of the circumference is actually the length of the radius multiplied by $2$. In our case, we already know what the diameter is, so all we have to do to find the length of the radius is to divide the diameter by $2$. By dividing it, we get that the radius of the circumference equals $10 cm$ $(20/2)$ .
2. As we have already said, to calculate the perimeter of the circumference we must multiply the value of $π$ l the length of the radius by $2$ (or use the value of the diameter directly instead of multiplying the length of the radius by $2$). The value of $π$ is $3,14$.

We obtain that:

$P=2\times R\timesπ=2\times10\times3,14=62,8$

The perimeter of the circumference is $62,8 cm$.To calculate the area of the circle, we must raise the length of the radius (obtained in a. above) of the circumference to the square and then multiply the result obtained by the value of $π$.

Thus, we obtain:

$S=π\times R\times R=3,14\times10\times10=314$

1. The area of the circle is $314 cm²$.

Do you know what the answer is?

## Circumference exercises:

### Exercise 1

Assignment:

Given the circumference of the figure

The diameter of the circle is $13$,

Solution

It is known that the diameter of the circle is twice its radius, i.e. it is possible to know the radius of the circle in the figure.

$13:2=6.5$

Tofind the area of the circle, we replace the data we have in the formula for the calculation of the circle

$A=\pi\times R²$

We replace the data we have:

$A=\pi\times6.5²$

$A=\pi42.25$

$pi42.25$

### Exercise 2

Request

Given an equilateral triangle in a circle

What is the areaof the circle?

Solution

Recall first the theorem that a circumferential angle that is inclined about the diameter is equal to 90 degrees.

That is, the triangle inside the circle is a right triangle and isosceles, so we can use the Pythagorean formula.

We replace the data we have with the Pythagorean formula

$X=Diámetro$

$(\sqrt{2})²+(\sqrt{2})²=X²$

The root cancels the power and therefore we obtain that

$2+2=X²$

That is

$X²=4$

The root of $4$ is $2$ and therefore the diameter is equal to $2$ and the radius is equal to half of the diameter and therefore it is equal to $1$

Therefore we obtain that the diameter is equal to $2$

And the radius is $1$

Then we add the formula for the area of the circle

$π$

### Exercise 3

Given the semicircle:

Consigna

Calculate its area

Solution

Since we know that it is a semicircle we can conclude that the base of the semicircle is the diameter.

We know that the diameter is twice the radius and therefore we can know the radius of the circle.

$Diámetro = 14$

$14:2=7$

$Radio = 7$

The formula for calculating the area of the circumference is

$A=\pi\times R²$

We replace the data in the formula

$A=\pi\times 7²$

$A=\pi\times 49$

Since the formula was the area of the semicircle we divide the area of the circle by $2$ and get the answer.

$\pi49:2=24.5\pi$

$\pi49:2=24.5\pi$

### Exercise 4

Request

Given the circumference of the figure

Given that the radius is equal to $6$,

What is the circumference?

Solution

The radius of the circle is $r=6$

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

Replace by $r=6$

We obtain

$2\cdot\pi\cdot6$

$12\pi$

$12\pi$

Do you think you will be able to solve it?

### Exercise 5

Request

Given the circumference of the figure.

Given that the radius is equal to $\frac{2}{3}$,

What is the circumference?

Solution

The radius of the circle $r=\frac{2}{3}$

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

Replace by $r=\frac{2}{3}$

We obtain

$2\cdot\pi\cdot\frac{2}{3}=$

$\frac{4}{3}\pi$

$\frac{4}{3}\pi$

### Exercise 6

Request

Given the circumference of the figure.

Given that the radius is equal to $5$,

What is the circumference?

Solution

Diameter of the circumference

$2r=5$

We divide by $2$

$r=\frac{5}{2}=2\frac{1}{2}$

Therefore the radius of the circle is $r=2\frac{1}{2}$

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

Replace by $r=2\frac{1}{2}$

We obtain

$2\cdot\pi\cdot2\frac{1}{2}=$

$2\cdot2\frac{1}{2}\cdot\pi=$

$5\pi$

$5\pi$

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## examples with solutions for circle

### Exercise #1

Given the circle whose diameter is 7 cm

### Step-by-Step Solution

First, let's remember the formula for the area of a circle:

$\pi r^2$

In the question, we are given the diameter of the circle, but we need the radius.

It is known that the radius is actually half of the diameter, therefore:

$r=7:2=3.5$

We replace in the formula

$\pi3.5^2=12.25\pi$

$12.25\pi$ cm².

### 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 replace the data in the formula:

$P=2\times6\times\pi$

$P=12\pi$

$12\pi$

### Exercise #3

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

What is its area?

### Step-by-Step Solution

Remember that the formula for the area of a circle is

πR²

We replace the data we know:

π3²

π9

$9\pi$ cm²

### Exercise #4

Look at the circle in the figure:

The radius is equal to 7.

What is the area of the circle?

### Step-by-Step Solution

Remember that the formula for the area of a circle is

πR²

We replace the data we know:

π7²

π49

49π

### Exercise #5

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:

$2\pi r$