# The Center of a Circle

🏆Practice parts of the circle

The center of the circumference belongs to subtopics that make up the topic of the circumference and the circle. We use the concept of the center of the circumference to define the circumference itself, as well as to calculate the radius and diameter of each given circumference.

The center of the circumference, as its name indicates, is a point located in the center of the circumference. It is usually customary to mark this point with the letter O. Indeed, this point is at the same distance from each of the points that make up the circumference.

## Test yourself on parts of the circle!

M is the center of the circle.

Perhaps $$MF=MC$$

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Below are some examples of different circumferences:

Each of them has a circumference center:

## Exercises on the center of the circle:

### Exercise 1

Assignment

Given the circle in the figure, $O$ is the center,

What is the circumference?

Solution

The radius is a straight line that connects the center of the circle and its circumference according to the figure is $4\text{ cm}$

Circumference formula

$2\pi r$

We replace accordingly based on the data

$2\pi\cdot4=8\pi$

$8\pi$

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### Exercise 2

Assignment

Given the circle with center $O$

Is it possible to calculate its area?

Solution

The center of the circle is $O$

That is, the given line is the diameter.

Diameter = Radius multiplied by 2

$2r=10$

$r=5$

We use the formula to calculate the area

$A=\pi r^2=$

$\pi5^2=25\pi$

Yes, its area is $25\pi$

### Exercise 3

Assignment

Given two circles whose center is located at the same point $O$

Given the measurements of the figure

What is the area of the orange shape?

Solution

Dotted area $A$

Area of the large circle $A_1$

Area of the small circle $A_2$

$A=A_1-A_2$

$A_1=\pi r_1^2$

$r_1=2+2=4$

$A_1=\pi4^2=16\pi$

$A_2=\pi r^2$

$r_2=2$

$A_2=\pi2^2=4\pi$

$A=A_1-A_2=16\pi-4\pi=12\pi$

$12\pi$

Do you know what the answer is?

### Exercise 4

Assignment

Given the circle center $O$

Inside the circle, there is a square

What is the area of the combined white parts?

We replace

$\pi=3.14$

Solution

$A_1=\pi r^2$

Diameter $=9$

Therefore, the radius is equal $4.5$

$A_1=\pi\cdot(4.5)^2=20.25\pi$

$\frac{(diagonal\cdot diagonal)}{2}=A_2$

The formula for the area of a rhombus (the square is also a rhombus). In the square, the diagonals are equal and therefore all the diagonals are $9\operatorname{cm}$

$A_2=\frac{9\cdot9}{2}=\frac{81}{2}=40.5$

$A=20.25\pi-40.5=$

$=20.25\cdot3.14-40.5=$

$23.085\operatorname{cm}²$

$23.085\operatorname{cm}²$

### Exercise 5

Assignment

The trapezoid $ABCD$ is inside the circle, whose center $O$

The area of the circle is $16\pi\operatorname{cm}²$

What is the area of the trapezoid?

Solution

$A_o=\pi r^2=16\pi$

We cancel out the 2 pi and take the square root

$r=\sqrt{16}=4$

$DC=DO+OC=$

$R+R=2R=$

$2\cdot4=8$

$ABCD=\frac{(AB+CD)EO}{2}=$

$\frac{(5+8)3.5}{2}=$

$\frac{13\cdot3.5}{2}=22.75$

$22.75\operatorname{cm}²$

## What is the center of a circle?

The center of a circle is the midpoint that is at the same distance from the circumference to that point, it is exactly in the middle of the circumference.

## What is the diameter of a circle?

It is the line that touches the circumference from end to end but passes through the center, as shown in the following image.

Do you think you will be able to solve it?

## What are some elements of the circumference?

The circumference has some lines, which are presented in the image and let's define each one of them:

C (Center): It is the point that is at the center of the circumference

D (Diameter): It is the line that passes through the midpoint of the circumference, that is, it passes through the center and touches the circumference from end to end.

R (Radius): It is half of the diameter, and this line only touches the center at one point of the circumference.

CU (Chord): It is the line that touches the circumference from end to end but does not necessarily pass through the center.

S (Secant): Line that crosses the circumference, as shown in the image:

## What happens if the radius is equal to zero?

If the radius in this case is zero, then there is no circumference, since as we mentioned the radius is the line that goes from the center of the circumference to any point on it, and because in this case the radius equals zero, we are not drawing any line and therefore no circle.

## examples with solutions for the center of a circle

### Exercise #1

There are only 4 radii in a circle.

### Step-by-Step Solution

A radius is a straight line that connects the center of the circle with a point on the circle itself.

False

### Exercise #2

M is the center of the circle.

Perhaps $MF=MC$

Yes

### Exercise #3

M is the center of the circle.

In the figure we observe 3 diameters?

No

### Exercise #4

M is the center of the circle.

Perhaps $AB=CD$

No

### Exercise #5

Is there sufficient data to determine that

$GH=AB$