Diameter

If this article interests you, you may also be interested in the following articles:

• The circumference
• The center of the circumference
• Circle
• Radius
• Pi
• Perimeter of the circle
• 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

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

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 $2\pi$

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

Reduce by $pi$

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

$8=r$

Answer

Diameter = Radius multiplied by

$2$

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 $\frac{1}{8}$

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

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

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

$198.7\operatorname{cm}²$

Answer

$198.7\operatorname{cm}²$

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.

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

Answer

$83.11\operatorname{cm}²$

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

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

Answer

$\frac{49}{12}r$

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

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

$16=x^2$

$4=x$

Answer

$4\operatorname{cm}$

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 $A=\pi r^2$

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

Answer

$4\pi$

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:

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

$D=2r$

Task

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

$D=6.76\operatorname{cm}$

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:

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

Task

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

$D=7.9\operatorname{cm}$

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