Radius

🏆Practice parts of the circle

The radius is one of the many elements that exist in a circle. The radius is a segment that connects the center of the circle with any point located on the circle itself. Each circle has an infinite number of radii and their length is exactly the same, that is, they are identical.

The radius is used to calculate the diameter and perimeter of the circle, it is also used to obtain the area of the circle.

Below are several examples of different circumferences.

The colored parts are, in fact, some radii painted on each circumference:

The colored parts are, in fact, some painted radii on the circumference:

Radius

Radius_of_a_circle.2

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Test yourself on parts of the circle!

einstein

M is the center of the circle.

Perhaps \( MF=MC \)

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In this article we will learn what the radius is and we will see how we can use it to calculate the perimeter of the circle and the area of the circle.

The radius is a segment that connects the center of the circle with any point located on the circle itself. We will illustrate it with a graph

Radio 1

Every circle has a center point. In the following illustration it is marked with the letter O. Now we will draw a line from the center point to any other point on the circle.

Radio 2

This line is the radius of the circle, usually denoted by the letter R R uppercase or r r lowercase. We can draw an infinite number of radii on each circle and they will all be of identical length.

P6 - Radius

For example, on this circle we have drawn three radii. All the radii of the circle have the same length. That is, the radius of a circle has a fixed length.


Diameter

The diameter of the circumference is the chord that passes exactly through the center and is usually denoted by the letter D.

For example:

Radio 6

The length of the diameter is equal to twice the length of the radius. Can you understand why? We can imagine that the diameter is composed of two radii.


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Circumference

With the length of the radius we can calculate the perimeter of the circle and the area of the circle. Just for that we have the formulas that will help us to do it.

We will mark the perimeter of the circle with the letter P P . The formula to calculate the perimeter of the circle is:

C=2πr C=2πr

Let's explain it in words: the circumference equals 2 2 multiplied by the number PI, multiplied by the radius. Recall that the value of PI(which is detailed in other articles), is approximately equivalent to 3.14 3.14


Here are some examples:

Example 1

Given a circle, knowing that its radius measures 3 3 cm.

Radio 7

What is the perimeter of the circumference?

Solution:

Let's write it down

R=3 R=3

Now let's remember the formula we just learned to calculate the perimeter:

P=2πr P=2πr

Let's put in the formula the parameters and we will get:

P=2×3.14×3 P=2\times3.14\times3

P=18.84 P=18.84 cm

Thus, we have based on the length of the radius to find the perimeter.

Area of the circle

With the radius we can also calculate the area of the circle, usually denoted by the letter A A . Just for that we have the following formula:

A=πr2 A=πr^2

Let's explain it in words: the area of the circle equals PI times the radius squared.


Do you know what the answer is?

Example 2

Given a circle with a radius of 4 4 cm.

Radio 8

What is the area of the circle?

Solution:

Let's mark the data:
R=4 R = 4 cm.

Now let's remember the formula to calculate the area of a circle:

A=πr2 A=πr^2

and let's set the parameters.

A=3.14×42 A=3.14\times4^2

A=3.14×16 A=3.14\times16

A=50.24 A=50.24

That is, we have arrived at that the area of the circle is 50.24 50.24 cm².

Pay attention to the units of measurement. The length of the radius is given in cm, but it is raised to the power of two and, therefore, the area is measured in cm² (cm squared).


Radius exercises

Exercise 1

Request

Given the circle in the figure

The radius of the circle is equal to: 9.5 9.5

What is its circumference?

Exercise 1- Assignment Given the circle in the figure

The radius of the circle is

r=912 r=9\frac{1}{2}

We use the formula of the circumference

2πr 2\pi r

We replace accordingly and get

2π912=19π 2\cdot\pi\cdot9\frac{1}{2}=19\pi

Answer

19π 19\pi


Check your understanding

Exercise 2

Problem

Given that the circumference is equal to: 8 8

What is the length of the radius of the circle?

Solution

According to the data 2πr=8 2\pi r=8

We divide on both sides by 2π 2\pi

2πr2π=82π \frac{2\pi r}{2\pi}=\frac{8}{2\pi}

Simplify and get

r=4π r=\frac{4}{\pi}

Answer

r=4π r=\frac{4}{\pi}


Exercise 3

Question

The radius of the circle is 4cm 4cm centimeters.

The length of the side of the square is 8cm 8cm centimeters.

In which shape is there a larger perimeter?

Solution

The circumference is: 2πr 2\pi r

We replace the data accordingly.

2π4=8π 2\cdot\pi\cdot4=8\pi

8π=83.14=25.12 8\pi=8\cdot3.14=25.12

The perimeter of the square is equal to 4a 4a

48=32 4\cdot8=32

Answer

Square


Do you think you will be able to solve it?

Exercise 4

Question

Given that the circumference of the circle is equal to 16 16

What is the length of the radius of the circle?

Solution

2πr=16 2\pi r=16

We divide the two sides by: 2π 2\pi

We obtain

2πr2π=162π \frac{2\pi r}{2\pi}=\frac{16}{2\pi}

Reduce 2π 2\pi

r=8r r=\frac{8}{r}

Answer

r=8r r=\frac{8}{r}


Exercise 5

Question

Given the circle in the figure

The radius of the circle is equal to: 14 \frac{1}{4}

What is the circumference?

The radius of the circle is equal to 1-4

Solution

The radius of the circle is equal to r=14 r=\frac{1}{4}

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

Replace accordingly

2π14=π2 2\cdot\pi\cdot\frac{1}{4}=\frac{\pi}{2}

Answer

π2 \frac{\pi}{2}


Test your knowledge

Review questions

What does radius mean in a circle?

Recall that the radius of a circle is a line segment that passes from the center of the circle and touches one of the points of the circumference, and is half the diameter. Let's look at the following image to see the radius of the circle.

Center of the circle and Radius


What is the radius of a circle 10 cm in diameter?

The radius is half of the diameter or we can say that the diameter is twice the radius, therefore if the diameter is equal to 10 cm 10\text{ cm} , then half will be the radius and therefore

r=5 cm r=5\text{ cm}


Do you know what the answer is?

What is the formula for calculating the perimeter and area of a circle knowing the radius and diameter?

The formula for finding the perimeter of a circle (the circumference) is the following:

P=2πr P=2\pi r

We can express it as two times the number pi times the radius.

Since the diameter is twice the radius we can also write this formula as:

P=πD P=\pi D

We can read the above formula as the perimeter is equal to pi times the diameter.

Now to calculate the area of a circle which is also known as the surface area we have the following formula:

A=πr2 A=\pi r^2


How to get the radius of a circle?

If we want to know the radius of a circle and we know the circumference or area we can use the formulas mentioned above,

For example if we know the circumference then we use the formula

P=2πr P=2\pi r

And from this formula we isolate the radius, dividing it all by 2π 2\pi , leaving the following form:

P2π=2πr2π \frac{P}{2\pi}=\frac{2\pi r}{2\pi}

Simplifying the formula:

P2π=r \frac{P}{2\pi}=r

In a general way for any circumference:

r=P2π r=\frac{P}{2\pi}

Now if we know the surface and we want to obtain the radius we use the formula:

A=πr2 A=\pi r^2

We isolate the radius in the equation, dividing all by pi:

Aπ=πr2π \frac{A}{\pi}=\frac{\pi r^2}{\pi}

Aπ=r2 \frac{A}{\pi}=r^2

Now, take the square root of both sides.

Aπ=r2 \sqrt{\frac{A}{\pi}}=\sqrt{r^2}

Simplifying and rearranging, we get the general way to find the radius of any circle if we are given the surface area

r=Aπ r=\sqrt{\frac{A}{\pi}}


Example 1

In a circle with circumference equal to 12 cm 12\text{ cm} , calculate the radius:

From our formula

r=P2π r=\frac{P}{2\pi}

We just substitute the value of the circumference

r=122π r=\frac{12}{2\pi}

r=6π r=\frac{6}{\pi}

Answer

r=6π r=\frac{6}{\pi}


Example 2

Question

A circle has an area of 64cm2 64\operatorname{cm}^2 , calculate the radius of the circumference:

Solution

In this case, the data we know is the area therefore from our formulas we have:

r=Aπ r=\sqrt{\frac{A}{\pi}}

Substituting the area:

r=64cm2π r=\sqrt{\frac{64\operatorname{cm}^2}{\pi}}

Find the square root:

r=64cm2π r=\frac{\sqrt{64\operatorname{cm}^2}}{\sqrt{\pi}}

r=8 cmπ r=\frac{8\text{ cm}}{\sqrt{\pi}}

Result:

r=8 cmπ r=\frac{8\text{ cm}}{\sqrt{\pi}}


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