# Diameter

🏆Practice parts of the circle

A diameter is a section that connects two points that lie on the circumference, that passes through the center of the circle. The diameter is actually twice the radius.

As in the case of the radius, as well as in the case of the diameter, there are an infinite number of diameters on the circumference, and all are identical in length.

Below is an example of a circle with several diameters marked in different colors.

## Test yourself on parts of the circle!

M is the center of the circle.

Perhaps $$MF=MC$$

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On the Tutorela blog you will find a variety of articles about mathematics.

## Diameter Exercises

### Exercise 1

Problem

Given the circle in the figure

What is its diameter?

Solution

We use the formula of the circumference

$2\pi r$

Replace the data

$16\pi=2\pi r$

Divide by $2\pi$

$\frac{16\pi}{2\pi}=r$

$\frac{16}{2}=r$

$8=r$

$2$

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### Exercise 2

Problem

What is the area of a slice of pizza whose diameter is $45\operatorname{cm}$ after dividing into $8$ slices?

Solution

Pizza divided by: $8$ slices

In other words, the area of a slice of pizza is $\frac{1}{8}$

$Apizza=\pi\cdot r²=\pi\cdot(\frac{diámetropizza}{2})²$

$\pi\cdot(\frac{45}{2})^2=506.25\pi$

$A=\frac{1}{8}\cdot506.25\pi$

$198.7\operatorname{cm}²$

$198.7\operatorname{cm}²$

### Exercise 3

Question

Given the parts of the circle shown in the figure (white)

The diameter of the circle is $11\operatorname{cm}$

How much is the area of the shaded parts together?

The area of the shaded parts is the area of the circle minus the two sections, one of which extends by an angle of $30°$ and the other by an angle of $15°$

In the same way we can look at the parts like this:

Then its area is the area of the circle minus the area of the section extended by an angle of $(45°)$

Or it is just a cut area extending by $(360°)$ degrees minus $(45°)$ degrees, i.e.,$(315°)$ degrees.

$A=\frac{315°}{360°}\cdot\pi\cdot(\frac{11}{2})^2=83.11$

$83.11\operatorname{cm}²$

Do you know what the answer is?

### Exercise 4

Question

In a circle, a cut is formed by an angle of $(120°)$ degrees.

Diameter of the angle $7\operatorname{cm}$

What is the dotted area?

Solution

The total circle has $360°$ degrees, $120°$ degrees is one third of $360°$ and therefore the area of the shape is equal to one third of the area of the circle.

We will use the formula for the area of the circle and replace accordingly.

$\pi r^2=\pi\cdot(\frac{7}{2})^2$

$\pi\cdot\frac{49}{4}$

We calculate the dotted area

$\frac{1}{3}\cdot\frac{49}{4}\cdot r=\frac{49}{12}r$

$\frac{49}{12}r$

### Exercise 5

Question

$ABCD$ is a rectangular trapezoid

Given that

Given that $AD$ is perpendicular to $CA$

$BC=X$

$AB=2X$

The area of the trapezoid is $\text{2}.5x^2$

The area of the circle that its diameter $AD$ is $16\pi$ $cm²$

Find a $x$

Solution

The area of $ABCD$ is equal to

$\text{ABCD}=\frac{(ab+dc)bc}{2}$

Replace accordingly

$2.5x^2=\frac{(2x+dc)x}{2}$

Multiply by $2$

$5x^2=(2x+dc)x$

Divide by $x$

$5x=2x+dc$

We go to $16\pi$ $cm²$ to the left section and keep the appropriate sign

$5x-2x=dc$

$3x=dc$

Calculate the triangle $\triangle ABC$

$ab^2+bc^2=ac^2$

Replace accordingly

$(2x)^2+x^2=ac^2$

$4x^2+x^2=5x^2=ac^2$

Take the root

$\sqrt{5}x=ac$

Calculate the triangle $\triangle ADC$

$ac^2+ad^2=dc^2$

Replace accordingly

$(\sqrt{5}x)^2+ad^2=(3x)^2$

$5x^2+ad^2=9x^2$

We move to the right to $5x$ and keep the appropriate sign

$ad^2=9x^2-5x^2$

$ad^2=4x^2$

We take the root

$ad=2x$

The area of the circle is equal to

$A=\pi\cdot(\frac{ad}{2})^2$

$16\pi=\pi\cdot(\frac{2x}{2})^2$

We reduce to $2$ and divide by pi

$16\pi=\pi\cdot(\frac{2x}{2})^2=\pi x^2$

$16=x^2$

$4=x$

$4\operatorname{cm}$

### Exercise 6

Question

Given the circle whose diameter $4\operatorname{cm}$

What is its area?

Solution

Diameter = Radius multiplied by $2$

That is to say

$2r=4$

Divide by $2$

$r=2$

Replace in the formula for the area of the circle $A=\pi r^2$

$A=\pi2^2=\pi\cdot4=4\pi$

$4\pi$

## Review questions

### What is the diameter of a circle?

The diameter of a circle is the straight line that passes through the center of the circle and touches from end to end of the circle, it is twice the radius.

Let's see the following image:

Do you think you will be able to solve it?

### How do we get the diameter of a circle with the area?

When we know the area or surface of a circle and we want to know the diameter of the circle we can use the area formula:

$A=\pi r^2$

From the above formula we know the surface area and the value of $\pi=3.14$, therefore we can find the radius as follows:

$\frac{A}{\pi}=\frac{\pi r^2}{\pi}$

$\frac{A}{\pi}=r^2$

We clear again, taking the root of both sides.

$\sqrt{\frac{A}{\pi}}=\sqrt{r^2}$

Simplifying we get the general way to know the radius of any circle knowing the area

$r=\sqrt{\frac{A}{\pi}}$

Now knowing the radius, with this we can also know the diameter, since the diameter is twice the radius, then we can write this as follows

$D=2r$

#### Example

Determine the diameter of the circle with area equal to $36\operatorname{cm}^2$

Solution

Since we know the area we are going to use the formula $A=\pi r^2$, in this case we want to know the radius so the simplified formula is

$r=\sqrt{\frac{A}{\pi}}$

Substituting it is as follows

$r=\sqrt{\frac{36\operatorname{cm}^2}{\pi}}$

$r=\sqrt{\frac{36\operatorname{cm}^2}{3.14}}$

$r=\sqrt{11.46\operatorname{cm}^2}$

$r=3.38\operatorname{cm}$

Now that we know the radius we can know the diameter since we know that the diameter is twice the radius.

$D=2r$

$D=2\left(3.38\operatorname{cm}\right)=6.76\operatorname{cm}$

Result

$D=6.76\operatorname{cm}$

### How to calculate the diameter of a circle knowing its circumference?

When we know the circumference we can use any of the two formulas of the diameter:

$P=2\pi r$

$P=\pi D$

In this case we will use the second formula, since this expressed the diameter, clearing the diameter dividing all by $pi$ we get.

$\frac{P}{\pi}=\frac{\pi D}{\pi}$

Simplifying we obtain:

$D=\frac{P}{\pi}$

#### Example

Find the diameter of a circle of $25 m$

Solution

From the above we can use

$D=\frac{P}{\pi}$

Substituting we have

$D=\frac{25\operatorname{cm}}{\pi}=\frac{25\operatorname{cm}}{3.14}=7.9\operatorname{cm}$

Result

$D=7.9\operatorname{cm}$

### What is the diameter of the Earth?

The diameter of the earth is about $12,756\text{ km}$

Do you know what the answer is?

## Examples with solutions for Diameter

### Exercise #1

M is the center of the circle.

Perhaps $AB=CD$

### Step-by-Step Solution

CD is a diameter, since it passes through the center of the circle, meaning it is the longest segment in the circle.

AB does not pass through the center of the circle and is not a diameter, therefore it is necessarily shorter.

Therefore:

$AB\ne CD$

No

### Exercise #2

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 #3

Which figure shows the radius of a circle?

### Step-by-Step Solution

It is a straight line connecting the center of the circle to a point located on the circle itself.

Therefore, the diagram that fits the definition is c.

In diagram a, the line does not pass through the center, and in diagram b, it is a diameter.

### Exercise #4

Which diagram shows a circle with a point marked in the circle and not on the circle?

### Step-by-Step Solution

The interpretation of "in a circle" is inside the circle.

In diagrams a'-d' the point is on the circle, and in diagram c' the point is outside the circle.

### Exercise #5

Is it possible that the circumference of a circle is 8 meters and its diameter is 4 meters?

### Step-by-Step Solution

To calculate, we will use the formula:

$\frac{P}{2r}=\pi$

Pi is the ratio between the circumference of the circle and the diameter of the circle.

The diameter is equal to 2 radii.

Let's substitute the given data into the formula:

$\frac{8}{4}=\pi$

$2\ne\pi$

Therefore, this situation is not possible.