Area of a circle

🏆Practice circle area

The area of the circle is, in fact, the surface that is "enclosed" within the perimeter of the circumference. It is calculated by raising the radius of the circumference RR to the second power and multiplying the result by -> π π . The area of the circle is usually denoted by the letter A A .

The formula to calculate the area of a circle is:

A=π×R×R A=\pi\times R\times R

A A -> area of the circle
π>PI=3.14 \pi–>PI=3.14
R R -> Radius of the circumference

In problems that include the radius - We will use the radius in the formula.
In problems that include the diameter - We will divide it by 2 2 to obtain the radius and, only then, place the radius in the formula.
In problems that include the area and ask to find the radius - We will place the area in the formula and find the radius.

A1 - The formula to calculate the area of a circle

A=π×R×R A=π\times R\times R

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Test yourself on circle area!

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Find the area of the circle according to the drawing.

222

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Area of the circle

In this article, we will learn everything necessary about the area of a circle. First, we will know the formula to calculate the area of the circle and then, we will continue with questions on the topic that could appear in an exam and that you should know how to solve.
Shall we start?


What is the area of the circle?

The area of the circle represents "the interior" of the circumference. By finding the area, we actually obtain the surface "enclosed" within the circumference.

The area of the circle represents

The part painted in orange is the area of the circle.

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How is the area of a circle calculated?

To calculate the area of the circle we must use the following formula:

A=π×R2 A = π \times R^2

A A -> area of the circle
π>PI=3.14 \pi–>PI=3.14
R R -> Radius of the circumference

Note - Notice that in the formula R R is raised to the second power. Those who are not comfortable operating with powers can replace R2 R^2 with R×R R\times R .

After understanding what each part of the formula implies, we can write it down like this:

A1 - The formula to calculate the area of a circle

Area of the Circle - Solved Exercises

Now let's move on to practice with some questions about the area of the circle.

Questions about the area of the circle when the radius is given

These are the easiest questions about the area of the circle since they require nothing more than placing the data in the formula.
The radius is already given and all you have to do is place it in the formula.

Let's practice:
Calculate the area of the circle knowing that the radius is 4 4 cm.

Solution:
The given radius is 4 4 cm.
We will place it in the formula to find the area of the circle and we will obtain:

A=3.14×42 A=3.14\times4^2
A=3.14×16 A=3.14\times16
A=50.24 A=50.24

Answer:
The area of the circle is 50.24 50.24 cm2 cm^2 .


Do you know what the answer is?

Another exercise

Given the circle

represents the center of the circumference.
What is the area of the circle?

Solution:

A - The given radius is 3 cm

Observe: we know that M M represents the center of the circumference. Therefore, we deduce that the segment coming from there is the radius.
In the illustration, it is shown that the segment coming from M M measures 3 3 cm.
That is, the radius measures 3 3 cm.
We will place it in the formula and obtain:

A=3.14×32 A=3.14\times3^2
A=3.14×9 A=3.14\times9
A=28.26 A=28.26

Answer:
The area of the circle is A=28.26 A=28.26 cm2 cm^2 .


More practice:

The radius measures 5 5 cm.
What is the area of the circle?

Solution

The area of the circle can be calculated by placing the provided data:

A=π×R×R=3.14×5×5=78.5 A=\pi\times R\times R=3.14\times5\times5=78.5

Answer:

That is, the area of the circle is 78.5 78.5 cm2 cm^2 .


Questions about the area of the circle when the diameter is given

In this type of questions, we must carry out a preliminary step before placing the data in the formula.
The diameter of the circle is the chord that passes exactly through the center and is equal to two radii.
That is, to go from the diameter to the radius we must divide by 2 2

Mode of action:
First step - Divide the diameter by 2 2 .
Second step - Place in the formula.

Let's practice:
Given a circle whose diameter measures 10 10 cm.
What is the area of the circle?

Solution:
We realize that we were given the diameter, but we need the radius to place it in the formula. Therefore, we will divide the diameter by 2 2 and arrive at the radius.

We will obtain :
10:2=5 10:2=5

The radius measures 5 5 cm.
We will place it in the formula and obtain :

A=3.14×52 A=3.14\times5^2
A=3.14×25 A=3.14\times25
A=78.5 A=78.5

Answer:
The area of the circle is 78.5 78.5 cm2 cm^2 .


Check your understanding

Another question

Observe the illustration and calculate the area of the circle.
\( M \ represents the center of the circumference.

\( ab=8 \ cm.

8 cm - M represents the center of the circumference

Solution:
We know that M represents the center of the circumference, therefore, the chord ab shown in the illustration is the diameter of the circle.
We will divide it by 2 2 and arrive at the radius.
We obtain:
8:2=4 8:2=4

The radius measures 4 4 cm. We will place it in the formula for the area of the circle and obtain:

A=3.14×42 A=3.14\times4^2
A=3.14×16 A=3.14\times16
A=50.24 A=50.24

Answer:
The area of the circle is 50.24 50.24 cm2 cm^2 .


Problems in which we have the area and must find the radius

In this type of problems, we will place the given area within the formula for calculating the area of the circle and, in this way, we will find the radius.
Remember A A : represents the area of the circle.


Exercise:
The area of the circle is 153.86 153.86
Calculate the radius of the circle.

Solution:
We will place in the formula:
r2×3.14=153.86 r^2\times3.14=153.86
We divide by 3.14 3.14
49=r2 49=r^2
We will clear the root
r=7 r=7

Answer:

The radius measures 7 7 cm.


Explanations in a more visual way.


Examples and exercises with solutions for the area of a circle

Exercise #1

Given the circle whose diameter is 7 cm

What is your area?

777

Video Solution

Step-by-Step Solution

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

 πr2 \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 r=7:2=3.5

We replace in the formula

π3.52=12.25π \pi3.5^2=12.25\pi

Answer

12.25π 12.25\pi cm².

Exercise #2

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

What is its area?

333OOO

Video Solution

Step-by-Step Solution

Remember that the formula for the area of a circle is

πR²

 

We replace the data we know:

π3²

π9

 

Answer

9π 9\pi cm²

Exercise #3

Look at the circle in the figure:

777

The radius is equal to 7.

What is the area of the circle?

Video Solution

Step-by-Step Solution

Remember that the formula for the area of a circle is

πR²

 

We replace the data we know:

π7²

π49

Answer

49π

Exercise #4

A circle has an area of 25 cm².

What is its radius?

Video Solution

Step-by-Step Solution

Area of the circle:

S=πr2 S=\pi r^2

We replace the data we know:

25=πr2 25=\pi r^2

Divide by Pi:25π=r2 \frac{25}{\pi}=r^2

Extract the root:25π=r \sqrt{\frac{25}{\pi}}=r

5π=r \frac{5}{\sqrt{\pi}}=r

Answer

5π \frac{5}{\sqrt{\pi}} cm

Exercise #5

Look at the circle in the figure:

131313

The diameter of the circle is 13.

What is its area?

Video Solution

Step-by-Step Solution

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

S=πr2 S=\pi r^2

The problem gives us the diameter, and we know that the radius is half of the diameter therefore:

132=6.5 \frac{13}{2}=6.5

We replace in the formula and solve:

S=π×6.52 S=\pi\times6.5^2

S=42.25π S=42.25\pi

Answer

42.25π

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