The Center of a Circle

If you are interested in this article, you might also be interested in the following articles:

• The circumference
• Circle
• Radius
• Diameter
• Pi
• The perimeter of the circumference
• Circular area
• Arcs in a circle
• Chords in a circle
• Central angle in a circle
• Perpendicular to the chord from the center of the circle
• Inscribed angle in a circle

In the Tutorela blog, you will find a variety of articles about mathematics

Below are some examples of different circumferences:

Each of them has a circumference center:

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

$2\pi\cdot4=8\pi$

Answer

$8\pi$

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

$A=\pi r^2=$

$\pi5^2=25\pi$

Answer

Yes, its area is $25\pi$

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 $A_2$

$A=A_1-A_2$

$A_1=\pi r_1^2$

$r_1=2+2=4$

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

$A_2=\pi r^2$

$r_2=2$

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

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

Answer

$12\pi$

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= Radius multiplied by 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 $9\operatorname{cm}$

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

$A=20.25\pi-40.5=$

$=20.25\cdot3.14-40.5=$

$23.085\operatorname{cm}²$

Answer

$23.085\operatorname{cm}²$

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

$r=\sqrt{16}=4$

$DC=DO+OC=$

$R+R=2R=$

$2\cdot4=8$

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

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

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

Answer

$22.75\operatorname{cm}²$

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.

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

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:

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.