Mathematics for Junior High School

By Marsigit
Three-dimensional Curved Surfaces Shape

Basic Competency:

To identify the elements of cylinder, cone, and sphere.
To estimate the lateral area and volume of cylinder, cone, and sphere.
To solve the problem that is related with cylinder, cone, and sphere.

What will be learned with this chapter?

Cylinder
Cone
Sphere

Cylinder


Definition of Cylinder

Cylinder or tube is a three-dimensional shape consists of two circular bases of equal area that are in parallel planes, and are connected by a lateral surface that intersects the boundaries of the bases.

The Elements of Cylinder

The elements of the cylinder are as follows:

Cylinder has an upper side (cap) and under side (base) in form of congruent circle (equal in its form and its size).
AB is called the diameter of cylinder base.
PE, PA, and PB are called the radius of cylinder base.
BC and AD is called the height of cylinder
The surface which is perpendicular to the bases is called the lateral of cylinder.
Two circular bases
The cap, the base and the lateral surface is called the surface of cylinder.


The Nets of Cylinder

Pay attention to the Figure 2.2 (a) is a cylinder that the radius is r and the height is t. If the cylinder as in the Figure 2.2 (a) is sliced as long as its height line (along AD or BC) and along its curved sides (along its perimeter of circle base and cap) as in the Figure 2.2 (b) then you will get the net of the cylinder as it seemed in the Figure 2.2 (c).

Figure 2.2

The cylinder which has the radius r and the height t.
The cylinder that is sliced along its curved sides of the base and the cap and along its height of cylinder.
The nets of cylinder.

In the nets of cylinder, it shows that the upper side (cap) and under side (base) are circles which has a radius r, while the net of the curved side (cylinder lateral) is rectangle ABCD.

Surface Area of Cylinder

The surface of the cylinder consists of cylinder lateral, upper side (cap), and under side (base). The cylinder lateral side is formed by a rectangle with length 2πr and width t.
The following formulas of areas are usually used in the cylinder:

Example:
The radius of the base of the cylinder is 7 cm and the height is 10 cm. Determine:
the length of the the lateral net.
the area of lateral surface, and
the area of cylinder surface.

Solution:
The height (t) is 10 cm and the cylinder base radius (r) is 7 cm.
The length of the the lateral net = 2πr
= 2 x x 7 = 44
So, the length of the the lateral net is 44 cm.

The area of lateral surface = 2πr x t
= 44 x 10
= 440
So, the area of lateral surface is 440 cm².

The area of cylinder surface = 2πr(t + r )
= 44 x (10 + 7)
= 44 x 17
= 748
So, area of cylinder surface is 749 cm².

Exercise:
An un-caped cylinder of glass has a diameter of 7 cm and a height of 20 cm. Determine:
the area of lateral surface, and
the area of cylinder surface.
A water pipe in the form of cylinder has the radius of 2.1 cm and the length of 28 cm. If the water pipe is hollow out at both its ends, determine the surface area of the pipe!
A clay-made flowerpot is in the form of cylinder. The base radius of the pot is 10 cm and the height is 20 cm. If the un-caped flowerpot will be painted on its lateral side and its base, determine the surface area should be painted!
A tart-cake for celebrating a birthday is in the form of cylinder with diameter 28 cm and height 8 cm. If it coated with chocolate, determine the surface area of the chocolate!
A can of milk is in the form of cylinder with diameter 7 cm and height 8 cm. The lateral side will be coated with paper containing information about the milk product. Determine the area of the paper!

5. The Volume of Cylinder

The way to identify the volume of cylinder is identical with that of the volume of a right prism. Consider the Figure 2.3!
Figure 2.3 A cylinder is a right prism with circular bases
If the base and the cap of regular prism as it shown in Figure 2.3 has a lot of facets so the form of the base and cap of the prism will close to the form of circle. The right prism with circular bases is called cylinder. Therefore, you will get the volume of the cylinder as the following:
The volume of the cylinder = the area of the base x the height of the cylinder
= the area of the circle x the height of the cylinder
= (πr²) x t
= πr²t.
So, the volume of the cylinder is πr²t, with r is radius of cylinder and t is height of cylinder.



Example:
Calculate the volume of a cylinder of radius 7 cm and height 20 cm!
Solution:
The radius of cylinder base (r) is 7 cm and the height of cylinder (t) is 20 cm. Therefore it goes in effect,
The volume of cylinder = πr²t
= x 7 x 7 x 20
= 22 x 7 x 20
= 3,080
So the volume of the cylinder is 3,080 cm³.

A cylinder is fully filled by 5,024 cm³ of water. The cylinder base radius is 10 cm.
Calculate the height of the water!
Solution:
The volume of cylinder is 5,024 cm³ and the radius (r) of the base is 10 cm. Let, the height of water is t cm, therefore it goes in effect,
Cylinder volume = πr²t
5,024 = 3.14 x 10² x t
5,024 = 3.14 x 100 x t
5,024 = 314 x t
t =
t = 16
So, the height of cylinder height is 16 cm.

Exercise:
A can of food that in form of cylinder has a height of 10 cm and a diameter of 7 cm. Determine the volume of the can!
A can of food that in form of cylinder has a radius of 10 cm. The can is loaded full by 11,000 cm³ of water. Determine the height of the cylinder!
A cylindrical drum of radius 30 cm and height 100 cm is fully filled with kerosene. Determine the volume of kerosene in the drum!
The lateral surface of a cylindrical can of biscuit is covered by a piece of gift paper. Ani wants to know the volume of the biscuit in the can. Ani then spreads out the cover of the can and finds out that the length is 88 cm and the width is 30 cm.
Determine the radius of the can?
Determine the volume of biscuit in the can!
A cylindrical can of paint has a height of 25 cm and a volume of 7,850 cm³. Determine the radius of the paint can!
Cone
Definition of Cone
A Cone is three-dimensional curved surfaces which has circular base of radius r and one side surface of revolution in the form of a sector of a circle.

The Elements of Cone
The elements of cone are as following:
A Cone consists of curved side surface called the lateral and the base in the form of circle.
PA and PC are called the radius of cone base
BP is called the height of a cone.
BA and BC are called the slant height.
Slant height (s) is a line that is connecting the peak of the cone to the point on base perimeter.

Figure 2.4 The component of a cone.

he Nets of Cone
Consider the Figure 2.5! Figure 2.5 (a) is a cone that has a radius r and a slant height s. If the cone as shown in Figure 2.5 (a) is sliced along its slant height s and along the curved base side (along the perimeter of base circle) then we will get the net of the cone as shown in Figure 2.5 (b).

Figure 2.5
A cone that has a radius r and a slant height s.
Radius of cone.


In the nets of a cone we see that the side of the base is a circle of radius r and a curved side (lateral of cone) is a sector of circle ABC of radius s.

The Area of the Surface of Cone
The surface of the cone consists of cone lateral and cone base. The cone lateral area (the circle sector area ABC with radius s) can be determined by the following comparison:

(The area of Circle sector ABC)/(The area of big circle )=( The lenght of Small arch AC)/(The perimeter of big circle )

(The area of cone lateral )/πs²= 2πr/2πs

The area of cone lateral = (π^2 s^2 r)/πs= πsr.

The area of cone surface = The area of cone lateral + the area of cone base
= πrs + πr² = πr(s + r)


Example:
The radius of a cone base is 6 cm. If the cone height is 8 cm, determine:
the area of cone lateral, and
the area of cone surface.

Solution:
The height (s) of the cone slant can be determined as follows:
s=√(r²+t²)= √(6²+8²)= √(36+64)= √100=10

Hence,
The area of cone lateral = πrs
= 3.14 x 6 x (10 + 6)
=188.4
So, The area of cone lateral is 188.4 cm².

The area of cone surface = πr (s + r)
= 3.14 x 6 x (10 + 6)
= 18.84 x 16
= 301.44
So, the area of cone surface is 301.44 cm².

Exercise:
A cone has a base radius of 7 cm and slant height of 20 cm. Determine the cone surface area!
A farmer has a wizard’s hat with the slant height 28 cm. The hat is made of bamboo matting for the width of 1,232 cm². Determine diameter of the farmer hat!
A trumpet in the form of cone is made of carton. If the area of the carton to make the trumpet is 550 cm² and yield the trumpet slant height 25 cm, determine the long of the trumpet!
A building has a shape of cone with the diameter of 12 m and height of 8 m. Determine the lateral area of the building!
Mother will make a hat for younger brother. The hat is in the formed of cone that has a base diameter of 21 cm and height of 14 cm. Determine the area of the component to make the hat!

5. The Volume of Cone
Consider the following Figure 2.6!
Figure 2.6 A cone is the regular pyramid with a lot of numbers of facets.

If the base of regular pyramid as in Figure 2.6 has a lot of facets, hence the pallet form of the regular pyramid is almost seems like circle. Pyramid that its base is in the form of circle is called a cone.

The volume of the cone = 1/3 x cone base area x cone height.
= 1/3 x πr² x t


Example:
Calculate the volume of the cone that has a radius of 3 cm and slant height of 5 cm!

Solution:

The base radius (r) of the cone is 3 cm and the slant height (s) of the cone is 5 cm. The height of the cone is determined as following:

s² = r² + t²
t² = s² - r² = 5² - 3² = 25 – 9 = 16
t = √16
t = 4

Therefore,
Volume of cylinder = 1/3 πr²t
= 1/3 x 3.14 x 3² x 4
= 37. 68

So, the volume of the cone is 37. 68 cm³.


Exercise:
A cone with radius 9 cm and slant height 15 cm. Determine the volume of the cone!
Mrs. Tuti will make a ceremonial dish of yellow rice served in a cone shape. It has a height of 56 cm and base radius of 42 cm. Determine the volume of the dish made by Mrs. Tuti!
Let, the perimeter of a base circle of cone is 132 cm and a slant height is 35 cm. Determine the volume of the cone!
A truncated-cone has a height of 21 cm and a cap radius of 20 cm. Determine the volume of the cone!
A glass is in the form of a cone. The perimeter of the upper glass is 22 cm. If the height is 10 cm, determine the volume of the glass!


Sphere

The Definition of Sphere

Sphere is a three-dimensional shape described by the rotation of a semicircle about its diameter.



The Area of Sphere Surface

It is not like a cylinder and a cone that have curved edge, not also like a cone that has an angle point, sphere hasn’t curved edge and angle point. Sphere only has one curved surface called the surface of sphere.
The area of the surface of the sphere can be found by use the formula as follow:

The surface area of the sphere = 2 x the surface area of hemisphere
= 2 x (2 x the area of circle)
= 2 x (2 x πr²)
= 4πr².

Example:
The radius of the sphere is 10 cm. Determine the surface area of the sphere!
Solution:
The surface area of sphere = 4πr²
= 4 x 3.14 x 10²
= 12.56 x 100
= 1,256
So, the surface area of sphere is 1,256 cm². Figure of sphere.

Exercise:
A sphere with radius 7 cm. Determine the surface area of the sphere!
To cover the surface of a sphere it needs material of 1,386 cm². Determine the diameter of the sphere!
A bowl in the form of hemisphere. The perimeter of the upper edge of the bowl is 31.4 cm. Determine surface area of the bowl!
A building has a roof in the form of hemisphere with diameter 14 m. The roof is made of glass. If the price of the glass is Rp 500,000.00 per m², determine the cost of the entire roof surface!
The perimeter of the bisector of sphere is 50.24 cm. Determine the surface area of the sphere!

The Volume of a Sphere

Volume of sphere can be determined by the following formula:

The volume of sphere = 2 x thevolume of hemisphere.
= 2 x (2 x volume of cone)
= 4 x 1/3πr²t = 4/3πr²t
= 4/3πr³ (remember: height of cone (t) = radius of cone (r))



Example:
Calculate the volume of the sphere with radius 10 cm!
Solution:
The volume of sphere = 4/3πr³ = 4/3 x 3,140 = 4,186.67
So, the volume of the sphere is 4,186.67 cm³.

Exercise:
A sphere has a radius of 14 cm. Calculate the volume of the sphere!
Let, the surface area of the sphere is 616 cm². Determine:
the radius of sphere, and
the volume of sphere.
Let, the volume of the sphere is 288π cm³. Determine the diameter of the sphere!
The inside part of an unripe coconut contains full of coconut milk. After the coconut milk is poured, its volume is 1,437 1/3 cm³ (the unripe coconut is assumed to have a spherical shape). Determine the diameter of the coconut if the thickness of the coconut and its shell is 0.5 cm! (Use π = 22/( 7))
An orange is sliced athwartly in equal size. It found that the orange diameter is 7 cm (the orange is assumed as spherical shape). Determine the volume of the half of orange!


Exercise 2


Choose the Correct Answer of the following problems


A sardine can is in the form of cylinder. The can has a radius of 7 cm and a height of 10 cm. The volume of the can is….
1,550 cm³ c. 1,504 cm³
1,540 cm³ d. 1450 cm³

The lateral area of the cylinder with radius 10 cm and height 20 cm is…
2,356 cm² c. 1,265 cm²
1,356 cm² d. 1,256 cm²

A can of drink in the form of cylinder on which its lateral side will be covered by paper. After the paper is unfolded, the size of the paper has a length of 62.8 cm and a width of 12 cm. The volume of the can is…
6,378 cm³ c. 3,678 cm³
3, 768 cm³ d. 3,578 cm³

A cylindrical shape of a can is filled up by wall paint. The can has a diameter of 20 cm and a height of 19 cm. The can volume is…
5,696 cm³ c. 5,969 cm³
5,966 cm³ d. 5,996 cm³

The younger brother buys milk as much as 2,009.6 cm³. He looks for the can to save the milk. He found the can in form of cylinder with the height of 10 cm. He found that the can is exactly filled by the milk. The diameter of the can is…
20 cm c. 16 cm
18 cm d. 14 cm

A can of drink is in formed of cylinder. The drink of the drink has a diameter of 2.8 cm and a high of 10 cm. The volume of the drink is….
63.6 cm³ c. 61.6 cm³
62.6 cm³ d. 60.6 cm³

A drum of kerosene has a height of 100 cm. The drum can load a full of kerosene as much as 138,600 cm³. The diameter of the drum is…
21 cm c. 35 cm
28 cm d. 42 cm

A cone has a height of 28 cm and a circle base radius of 21 cm. The lateral area is….
38,808 cm² c. 3,210 cm²
12,936 cm² d. 2,310 cm²

My brother buys boiled peanut in a vendor. The seller wraps it with the conical shape of paper that has a cap radius of 5 cm and a height of 15 cm. The volume of the cone is…
235.5 cm³ c. 392.5 cm³
382.5 cm³ d. 1,177.5 cm³

Tono was born in 21st. Therefore, on his birthday for 14th, a ceremonial dish of yellow rice in the shape of cone has its diameter of 14 cm and height of 21 cm. The volume of the shape is….
3,324 cm³ c. 1,780 cm³
3,234 cm³ d. 1,078 cm³

Ani will make a hat in form of cone that has a circle base perimeter of 44 cm. If the slant height is 10 cm, so, the lateral area of Ani’s hat is…
1,540 cm² c. 440 cm²
513.33 cm² d. 220 cm²

The area of paper that as the lateral of the cone is 753.6 cm². The slant height is 20 cm. The radius of the cone base is…
8 cm c. 12 cm
10 cm d. 16 cm

A cone is fully filled with 2.198 dm³ of fried nut. If the diameter of the cone’s cap is 20 cm, so the height of the cone is…
10 cm c. 21 cm
20 cm d. 22 cm

Brother sliced the orange athwartly into two similar parts. The diameter of the orange is 7 cm (the orange is assumed to have a spherical form). The area of the orange peel is…

616 cm² c. 154 cm²
166 cm² d. 145 cm²

Mother buys a watermelon. The perimeter of the surrounding watermelon is 62.8 cm. (the watermelon is assumed as sphere). The volume of the watermelon is…
628 cm³ c. 2,093.33 cm³
1.256 cm³ d.4,186.67 cm³

A sphere has a radius of 9 cm. The volume of the sphere is…
339.12 cm³ c. 1,017.36 cm³
678.24 cm³ d. 3,052.08 cm³

The surface area of the sphere is 1,808.64 cm². The volume of the sphere is…
2,411.52 cm³ c. 7,234.56 cm³
4,823.04 cm³ d. 7,236.56 cm³

A ball is fully filled up with 1,4371/3 cm³ of sands. The diameter of the ball is…
7 cm c. 14 cm
12 cm d. 21 cm

A ball has a volume of 904.32 cm³. The surface area of the ball is…
2,1712.96 cm² c. 425.16 cm²
452.16 cm² d. 254.16 cm²
A ball has diameter 24 cm. The surface of the ball will be covered with decorative paper. The area of the paper is…
1,880.64 cm² c. 150.72 cm²
1,808.64 cm² d. 105.72 cm²



Solve the following problems

A gasholder is in the form of cylinder. It is fully filled up by 2,355 dm³ of kerosene. If the height of the gasholder is 300 cm, determine:
the diameter of the gasholder, and
the surface area of the gasholder.

Budi wants to make a ceremonial dish of yellow rice in the shape of cone (tumpeng) that has a height of 30 cm. If he wants the base area of the tumpeng is 616 cm²,
How long the radius of the tumpeng?
How big the volume of the tumpeng?

Brother buys pop corn in conic paper bag. If the volume of the pop corn is 314 cm³ and the cap diameter is 10 cm, calculate:
The height of the bag, and
The paper area that is the pop corn packer.

Andi has two globes made of glass. One of the globes has a diameter of 15 cm and the other has a diameter of 0.5 cm.
What the volume of the globe?
What the area of the glass to make the globe?