Figure 2.3

Figure 2.2

Figure 2.1

Figure 2.4

22..11

Total Surface Area

What are you going to learn?

to mention the elements of a cylinder.

to draw cylinder nets. to explain the formula of

the area of a cylinder’s surface.

to calculate the area of a cylinder’s surface.

to explain the formula of the volume of a cylinder.

to calculate the volume of a cylinder.

to calculate the height or radius of a cylinder if the volume is given.

Key Terms:

Observe the pictures of

some cans on the right .

What are the shapes of the

cans?

A cylinder has two circular

bases and a lateral face.

• cylinder • area of a cylinder’s

surface • area of a cylinder’s base • volume of a c

Here are the mathematical figures.

ylinder

A

B

A

B

If we cut the cylinder along the bases and along straight line AB in Figure 2.3 and we spread it on a plane, we will get a net as in Figure 2.4.

Mathematics for Junior High School Grade 9 / 33

For a better understanding, take a can of milk or any can with a label. If you cut the label as in Figure 2.5 and spread it on a plane, you will get a rectangular figure. The width of the rectangle is equal to the the height of the can and the length is the

circumference of the base. Figure 2.5

Now, how do we find the total surface area? Look at the net of the cylinder below. The total surface area can be found by adding all the three areas.

Total surface area = lateral area + 2 base area

= 2πrh + 2πr2

If the total surface area is called T, we have the formula for T:

Area of top base = πr2

Area of bottom base = πr2

Area of lateral face = 2πrh

Remember

π is a number indicating the comparison between a round of one circle (say K) with its diameter (say d)

π= dK

The approximate value of π is 7

22 or 3.14.

T = 2πrh + 2πr2, where r = radius of cylinder

h = height

Formula for the Total

Surface Area

34/ Student’s Book – Space Figures with Curved Surface

Find the smallest area of an alluminum sheet for making a can like the picture below. (Use π = 7

22 )

7 cm

11.5 cm

Volume of a Cylinder

What is the volume of a cylinder? We can use the formula of the volume of a prism.

(a) (b)

(c)

Figure 2.6 The volumes of regular prisms (a) and (b) are the base area (A) times the height (h). If the number of the sides increases to infinity, then the base will be a circle. So, the volume of the cylinder is:

V = A × h V = (π r2 ) × h

V = π r2 h, with r = radius of cylinder

h = height of cylinder

Formula for the Volume of a

Cylinder

Mathematics for Junior High School Grade 9 / 35

Real life Situation

In many birthday parties, we often serve a birthday cake like the picture the on the left. Mostly, the cake of this kind is circular in shape. The diameter of the cake is 10 cm and the height is 5 cm. Find the volume of the cake.

To celebrate his birthday Arry serves her guests a two-layer cake as seen on the picture on the left. The height of each layer is 7 cm. If the diameter of the bottom cake is 30 cm and the diamater of the top one is 25 cm, find the difference in volume of both layers.

1. Determine the total surface area and the volume of each of the following

cylinders.

a) b) c)

7 cm

10 cm 15 cm

14 cm

7 cm

12 cm

36/ Student’s Book – Space Figures with Curved Surface

2. The front side of a heavy machine is a cylindrical in

shape. The diameter of this part is 6 feet (foot/feet =

feet contracted into ft) and the length is 8 feet. How

wide is the surface of the cylinder? What is the

volume of the cylinder?

3. Someone wants to make a cylinder with the volume of 600 cm3. If the

radius of the base of the cylinder is 5 cm, what is the height of the cylinder? 4. If the volume of a cylinder is 135 π cm3 and the height is 15 cm, what is the

radius of the cylinder?

5. An oil refinery whose height is 32 m and whose diameter of the base side is 84 m

will be painted. What is the area of the refinery to be

painted? If one gallon of paint can be used to paint an

area of 325 m3, how many gallons of paint are

required?

6. The radius of a cylindrical tank is enlarged so that its radius is twice larger than before. Find how much larger is its volume after being enlarged.

7. The surface of a swimming pool has a circle in shape with circumference of

77 meters. Calculate how much water is needed to fill in the pool if the height is 1.2 meters.

Mathematics for Junior High School Grade 9 / 37