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Diameter Exercises Exercise 1 Problem
Given the circle in the figure
What is its diameter?
Solution
We use the formula of the circumference
2 π r 2\pi r 2 π r
Replace the data
16 π = 2 π r 16\pi=2\pi r 16 π = 2 π r
Divide by 2 π 2\pi 2 π
16 π 2 π = r \frac{16\pi}{2\pi}=r 2 π 16 π = r
Reduce by p i pi p i
16 2 = r \frac{16}{2}=r 2 16 = r
8 = r 8=r 8 = r
Answer
Diameter = Radius multiplied by
2 2 2
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Exercise 2 Problem
What is the area of a slice of pizza whose diameter is 45 cm 45\operatorname{cm} 45 cm after dividing into 8 8 8 slices?
Solution
Pizza divided by: 8 8 8 slices
In other words, the area of a slice of pizza is 1 8 \frac{1}{8} 8 1
A p i z z a = π ⋅ r 2 = π ⋅ ( d i a ˊ m e t r o p i z z a 2 ) 2 Apizza=\pi\cdot r²=\pi\cdot(\frac{diámetropizza}{2})² A p i zz a = π ⋅ r 2 = π ⋅ ( 2 d i a ˊ m e t ro p i zz a ) 2
π ⋅ ( 45 2 ) 2 = 506.25 π \pi\cdot(\frac{45}{2})^2=506.25\pi π ⋅ ( 2 45 ) 2 = 506.25 π
A = 1 8 ⋅ 506.25 π A=\frac{1}{8}\cdot506.25\pi A = 8 1 ⋅ 506.25 π
198.7 cm 2 198.7\operatorname{cm}² 198.7 cm 2
Answer
198.7 cm 2 198.7\operatorname{cm}² 198.7 cm 2
Exercise 3 Question
Given the parts of the circle shown in the figure (white)
The diameter of the circle is 11 cm 11\operatorname{cm} 11 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 ° 30° 30° and the other by an angle of 15 ° 15° 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 ° ) (45°) ( 45° )
Or it is just a cut area extending by ( 360 ° ) (360°) ( 360° ) degrees minus ( 45 ° ) (45°) ( 45° ) degrees, i.e.,( 315 ° ) (315°) ( 315° ) degrees.
A = 315 ° 360 ° ⋅ π ⋅ ( 11 2 ) 2 = 83.11 A=\frac{315°}{360°}\cdot\pi\cdot(\frac{11}{2})^2=83.11 A = 360° 315° ⋅ π ⋅ ( 2 11 ) 2 = 83.11
Answer
83.11 cm 2 83.11\operatorname{cm}² 83.11 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°) ( 120° ) degrees.
Diameter of the angle 7 cm 7\operatorname{cm} 7 cm
What is the dotted area?
Solution
The total circle has 360 ° 360° 360° degrees, 120 ° 120° 120° degrees is one third of 360 ° 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.
π r 2 = π ⋅ ( 7 2 ) 2 \pi r^2=\pi\cdot(\frac{7}{2})^2 π r 2 = π ⋅ ( 2 7 ) 2
π ⋅ 49 4 \pi\cdot\frac{49}{4} π ⋅ 4 49
We calculate the dotted area
1 3 ⋅ 49 4 ⋅ r = 49 12 r \frac{1}{3}\cdot\frac{49}{4}\cdot r=\frac{49}{12}r 3 1 ⋅ 4 49 ⋅ r = 12 49 r
Answer
49 12 r \frac{49}{12}r 12 49 r
Exercise 5 Question
A B C D ABCD A BC D is a rectangular trapezoid
Given that
Given that A D AD A D is perpendicular to C A CA C A
B C = X BC=X BC = X
A B = 2 X AB=2X A B = 2 X
The area of the trapezoid is 2 . 5 x 2 \text{2}.5x^2 2 .5 x 2
The area of the circle that its diameter A D AD A D is 16 π 16\pi 16 π c m 2 cm² c m 2
Find a x x x
Solution
The area of A B C D ABCD A BC D is equal to
ABCD = ( a b + d c ) b c 2 \text{ABCD}=\frac{(ab+dc)bc}{2} ABCD = 2 ( ab + d c ) b c
Replace accordingly
2.5 x 2 = ( 2 x + d c ) x 2 2.5x^2=\frac{(2x+dc)x}{2} 2.5 x 2 = 2 ( 2 x + d c ) x
Multiply by 2 2 2
5 x 2 = ( 2 x + d c ) x 5x^2=(2x+dc)x 5 x 2 = ( 2 x + d c ) x
Divide by x x x
5 x = 2 x + d c 5x=2x+dc 5 x = 2 x + d c
We go to 16 π 16\pi 16 π c m 2 cm² c m 2 to the left section and keep the appropriate sign
5 x − 2 x = d c 5x-2x=dc 5 x − 2 x = d c
3 x = d c 3x=dc 3 x = d c
Calculate the triangle △ A B C \triangle ABC △ A BC
a b 2 + b c 2 = a c 2 ab^2+bc^2=ac^2 a b 2 + b c 2 = a c 2
Replace accordingly
( 2 x ) 2 + x 2 = a c 2 (2x)^2+x^2=ac^2 ( 2 x ) 2 + x 2 = a c 2
4 x 2 + x 2 = 5 x 2 = a c 2 4x^2+x^2=5x^2=ac^2 4 x 2 + x 2 = 5 x 2 = a c 2
Take the root
5 x = a c \sqrt{5}x=ac 5 x = a c
Calculate the triangle △ A D C \triangle ADC △ A D C
a c 2 + a d 2 = d c 2 ac^2+ad^2=dc^2 a c 2 + a d 2 = d c 2
Replace accordingly
( 5 x ) 2 + a d 2 = ( 3 x ) 2 (\sqrt{5}x)^2+ad^2=(3x)^2 ( 5 x ) 2 + a d 2 = ( 3 x ) 2
5 x 2 + a d 2 = 9 x 2 5x^2+ad^2=9x^2 5 x 2 + a d 2 = 9 x 2
We move to the right to 5 x 5x 5 x and keep the appropriate sign
a d 2 = 9 x 2 − 5 x 2 ad^2=9x^2-5x^2 a d 2 = 9 x 2 − 5 x 2
a d 2 = 4 x 2 ad^2=4x^2 a d 2 = 4 x 2
We take the root
a d = 2 x ad=2x a d = 2 x
The area of the circle is equal to
A = π ⋅ ( a d 2 ) 2 A=\pi\cdot(\frac{ad}{2})^2 A = π ⋅ ( 2 a d ) 2
16 π = π ⋅ ( 2 x 2 ) 2 16\pi=\pi\cdot(\frac{2x}{2})^2 16 π = π ⋅ ( 2 2 x ) 2
We reduce to 2 2 2 and divide by pi
16 π = π ⋅ ( 2 x 2 ) 2 = π x 2 16\pi=\pi\cdot(\frac{2x}{2})^2=\pi x^2 16 π = π ⋅ ( 2 2 x ) 2 = π x 2
16 = x 2 16=x^2 16 = x 2
4 = x 4=x 4 = x
Answer
4 cm 4\operatorname{cm} 4 cm
Exercise 6 Question
Given the circle whose diameter 4 cm 4\operatorname{cm} 4 cm
What is its area?
Solution
Diameter = Radius multiplied by 2 2 2
That is to say
2 r = 4 2r=4 2 r = 4
Divide by 2 2 2
r = 2 r=2 r = 2
Replace in the formula for the area of the circle A = π r 2 A=\pi r^2 A = π r 2
A = π 2 2 = π ⋅ 4 = 4 π A=\pi2^2=\pi\cdot4=4\pi A = π 2 2 = π ⋅ 4 = 4 π
Answer
4 π 4\pi 4 π
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 = π r 2 A=\pi r^2 A = π r 2
From the above formula we know the surface area and the value of π = 3.14 \pi=3.14 π = 3.14 , therefore we can find the radius as follows:
A π = π r 2 π \frac{A}{\pi}=\frac{\pi r^2}{\pi} π A = π π r 2
A π = r 2 \frac{A}{\pi}=r^2 π A = r 2
We clear again, taking the root of both sides.
A π = r 2 \sqrt{\frac{A}{\pi}}=\sqrt{r^2} π A = r 2
Simplifying we get the general way to know the radius of any circle knowing the area
r = A π r=\sqrt{\frac{A}{\pi}} r = π A
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 = 2 r D=2r D = 2 r
Example Task
Determine the diameter of the circle with area equal to 36 cm 2 36\operatorname{cm}^2 36 cm 2
Solution
Since we know the area we are going to use the formula A = π r 2 A=\pi r^2 A = π r 2 , in this case we want to know the radius so the simplified formula is
r = A π r=\sqrt{\frac{A}{\pi}} r = π A
Substituting it is as follows
r = 36 cm 2 π r=\sqrt{\frac{36\operatorname{cm}^2}{\pi}} r = π 36 cm 2
r = 36 cm 2 3.14 r=\sqrt{\frac{36\operatorname{cm}^2}{3.14}} r = 3.14 36 cm 2
r = 11.46 cm 2 r=\sqrt{11.46\operatorname{cm}^2} r = 11.46 cm 2
r = 3.38 cm r=3.38\operatorname{cm} r = 3.38 cm
Now that we know the radius we can know the diameter since we know that the diameter is twice the radius.
D = 2 r D=2r D = 2 r
D = 2 ( 3.38 cm ) = 6.76 cm D=2\left(3.38\operatorname{cm}\right)=6.76\operatorname{cm} D = 2 ( 3.38 cm ) = 6.76 cm
Result
D = 6.76 cm D=6.76\operatorname{cm} D = 6.76 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 π r P=2\pi r P = 2 π r
P = π D P=\pi D P = π D
In this case we will use the second formula, since this expressed the diameter, clearing the diameter dividing all by p i pi p i we get.
P π = π D π \frac{P}{\pi}=\frac{\pi D}{\pi} π P = π π D
Simplifying we obtain:
D = P π D=\frac{P}{\pi} D = π P
Example Task
Find the diameter of a circle of 25 m 25 m 25 m
Solution
From the above we can use
D = P π D=\frac{P}{\pi} D = π P
Substituting we have
D = 25 cm π = 25 cm 3.14 = 7.9 cm D=\frac{25\operatorname{cm}}{\pi}=\frac{25\operatorname{cm}}{3.14}=7.9\operatorname{cm} D = π 25 cm = 3.14 25 cm = 7.9 cm
Result
D = 7.9 cm D=7.9\operatorname{cm} D = 7.9 cm
What is the diameter of the Earth? The diameter of the earth is about 12 , 756 km 12,756\text{ km} 12 , 756 km
Do you know what the answer is?
Examples with solutions for Diameter 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.
Therefore, the answer is incorrect, as there are infinite radii.
Answer Exercise #2 M is the center of the circle.
Perhaps A B = C D AB=CD A B = C D
M M M A A A B B B C C C D D D E E E F F F G G G H H H
Video Solution 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:
A B ≠ C D AB\ne CD A B = C D
Answer Exercise #3 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) and (d) the point is on the circle, while in diagram (c) the point is outside of the circle.
Answer Exercise #4 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.
Answer Exercise #5 Is it possible that the circumference of a circle is 8 meters and its diameter is 4 meters?
Video Solution Step-by-Step Solution To calculate, we will use the formula:
P 2 r = π \frac{P}{2r}=\pi 2 r P = π
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:
8 4 = π \frac{8}{4}=\pi 4 8 = π
2 ≠ π 2\ne\pi 2 = π
Therefore, this situation is not possible.
Answer