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.

Diameter

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Test yourself on parts of the circle!

einstein

There are only 4 radii in a circle.

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Diameter Exercises

Exercise 1

Problem

Given the circle in the figure

What is its diameter?

1- Given the circle in the figure, what is its diameter?

Solution

We use the formula of the circumference

2πr 2\pi r

Replace the data

16π=2πr 16\pi=2\pi r

Divide by 2π 2\pi

16π2π=r \frac{16\pi}{2\pi}=r

Reduce by pi pi

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

8=r 8=r

Answer

Diameter = Radius multiplied by

2 2


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Exercise 2

Problem

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

Solution

Pizza divided by: 8 8 slices

In other words, the area of a slice of pizza is 18 \frac{1}{8}

Apizza=πr2=π(diaˊmetropizza2)2 Apizza=\pi\cdot r²=\pi\cdot(\frac{diámetropizza}{2})²

π(452)2=506.25π \pi\cdot(\frac{45}{2})^2=506.25\pi

A=18506.25π A=\frac{1}{8}\cdot506.25\pi

198.7cm2 198.7\operatorname{cm}²

Answer

198.7cm2 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 11cm 11\operatorname{cm}

How much is the area of the shaded parts together?

Exercise 3 - Given the parts of the circle shown in the white figure

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°30° and the other by an angle of 15°15°

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

What is the area of the parts together?

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

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

A=315°360°π(112)2=83.11A=\frac{315°}{360°}\cdot\pi\cdot(\frac{11}{2})^2=83.11

Answer

83.11cm2 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°) (120°) degrees.

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

What is the dotted area?

In a circle, a cut is formed at an angle of 120 degrees.

Solution

The total circle has 360°360° degrees, 120°120° degrees is one third of 360°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.

πr2=π(72)2 \pi r^2=\pi\cdot(\frac{7}{2})^2

π494 \pi\cdot\frac{49}{4}

We calculate the dotted area

13494r=4912r \frac{1}{3}\cdot\frac{49}{4}\cdot r=\frac{49}{12}r

Answer

4912r \frac{49}{12}r


Exercise 5

Question

ABCD ABCD is a rectangular trapezoid

Given that

Given that ADAD is perpendicular to CACA

BC=X BC=X

AB=2X AB=2X

The area of the trapezoid is 2.5x2 \text{2}.5x^2

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

Find a x x

5- ABCD is a right-angled trapezoid Given that AD is perpendicular to CA

Solution

The area of ABCD ABCD is equal to

ABCD=(ab+dc)bc2 \text{ABCD}=\frac{(ab+dc)bc}{2}

Replace accordingly

2.5x2=(2x+dc)x2 2.5x^2=\frac{(2x+dc)x}{2}

Multiply by 2 2

5x2=(2x+dc)x 5x^2=(2x+dc)x

Divide by x x

5x=2x+dc 5x=2x+dc

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

5x2x=dc 5x-2x=dc

3x=dc 3x=dc

Calculate the triangle ABC \triangle ABC

ab2+bc2=ac2 ab^2+bc^2=ac^2

Replace accordingly

(2x)2+x2=ac2 (2x)^2+x^2=ac^2

4x2+x2=5x2=ac2 4x^2+x^2=5x^2=ac^2

Take the root

5x=ac \sqrt{5}x=ac

Calculate the triangle ADC \triangle ADC

ac2+ad2=dc2 ac^2+ad^2=dc^2

Replace accordingly

(5x)2+ad2=(3x)2 (\sqrt{5}x)^2+ad^2=(3x)^2

5x2+ad2=9x2 5x^2+ad^2=9x^2

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

ad2=9x25x2 ad^2=9x^2-5x^2

ad2=4x2 ad^2=4x^2

We take the root

ad=2x ad=2x

The area of the circle is equal to

A=π(ad2)2 A=\pi\cdot(\frac{ad}{2})^2

16π=π(2x2)2 16\pi=\pi\cdot(\frac{2x}{2})^2

We reduce to 2 2 and divide by pi

16π=π(2x2)2=πx2 16\pi=\pi\cdot(\frac{2x}{2})^2=\pi x^2

16=x2 16=x^2

4=x 4=x

Answer

4cm 4\operatorname{cm}


Check your understanding

Exercise 6

Question

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

What is its area?

Exercise 6 - Given the circle whose diameter 4

Solution

Diameter = Radius multiplied by 2 2

That is to say

2r=4 2r=4

Divide by 2 2

r=2 r=2

Replace in the formula for the area of the circle A=πr2 A=\pi r^2

A=π22=π4=4π A=\pi2^2=\pi\cdot4=4\pi

Answer

4π 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=πr2 A=\pi r^2

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

Aπ=πr2π \frac{A}{\pi}=\frac{\pi r^2}{\pi}

Aπ=r2 \frac{A}{\pi}=r^2

We clear again, taking the root of both sides.

Aπ=r2 \sqrt{\frac{A}{\pi}}=\sqrt{r^2}

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

r=Aπ 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 D=2r

Example

Task

Determine the diameter of the circle with area equal to 36cm2 36\operatorname{cm}^2

Solution

Since we know the area we are going to use the formula A=πr2 A=\pi r^2 , in this case we want to know the radius so the simplified formula is

r=Aπ r=\sqrt{\frac{A}{\pi}}

Substituting it is as follows

r=36cm2π r=\sqrt{\frac{36\operatorname{cm}^2}{\pi}}

r=36cm23.14 r=\sqrt{\frac{36\operatorname{cm}^2}{3.14}}

r=11.46cm2 r=\sqrt{11.46\operatorname{cm}^2}

r=3.38cm 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=2r

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

Result

D=6.76cm 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=2πr P=2\pi r

P=πD P=\pi D

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

Pπ=πDπ \frac{P}{\pi}=\frac{\pi D}{\pi}

Simplifying we obtain:

D=Pπ D=\frac{P}{\pi}

Example

Task

Find the diameter of a circle of 25m 25 m

Solution

From the above we can use

D=Pπ D=\frac{P}{\pi}

Substituting we have

D=25cmπ=25cm3.14=7.9cm D=\frac{25\operatorname{cm}}{\pi}=\frac{25\operatorname{cm}}{3.14}=7.9\operatorname{cm}

Result

D=7.9cm D=7.9\operatorname{cm}


Test your knowledge

What is the diameter of the Earth?

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


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