Math 9th grade
LEARNING UNIT
Discover measurements
based on geometrical
shape
S/K
LEARNING OBJECT
To solve problems related with cylindrical
shapes.
SCO: Describes cylindrical shapes present in their
surroundings.
 Identifies the lateral face, the bases and the
height of a cylinder.
 Builds the concept of a cylinder.
 Recognizes the shape of an oblique cylinder
 Identifies the difference between a right
cylinder and an oblique cylinder
 Recognizes cylindric sections.
SCO: Recognizes the area of the surface of cylinders.
 Recognizes the figures that make up the surface
of a cylinder.
 Calculates the area of the bases of a cylinder.
 Calculates the area of the lateral face of a
cylinder.
 Calculates the total area of the surface of a
cylinder.
 Builds cylinders from the area measurements of
the surface.
 Finds the expense of material in the
construction of a cylinder by identifying the
surface area.
 Identifies the figures contained in the surface of
oblique cylinders.
 Expresses and supports procedures and
strategies for area calculation.
 Recognizes the figures contained in the surface
of right cylinders.
SCO: Recognizes the volume of a cylinder.
 Interprets the volume measurement of a
geometrical shape.
 Identifies the height of a right cylinder.
 Relates the height of a cylinder with the area of
the bases when calculating volume.
 Establishes strategies to determine the volume
of an oblique cylinder.
 Recognizes the formula to find the volume of a
cylinder.
 Calculates the volume of cylinders.
 Establishes strategies to find the volume of a
cylindric section.
Language
English
Socio cultural context of Colombia
the LO
Curricular axis
Spatial thinking, numerical reasoning.
Standard competencies
 Establish valid calculation procedures to find the
area of flat sections and the volume of solid
objects.
 Select and use methods and instruments to
measure lengths, surface areas, volumes and
angles with appropriate precision.
Background Knowledge Represents cubes, crates, cones, cylinders, prisms,
pyramids, in two dimensions.
Basic Learning
To know the formulae to calculate surface areas and
Rights
the volume of cylinders and prisms.
English Review topic
WH- Questions, YES/NO Questions
Vocabulary box
 Straight cylinder: Round geometrical object
formed by a lateral curved surface and two
parallel round planes that form its bases.
 Oblique cylinder: Cylinder whose generatrix forms
an angle different from 90° in relation to the plane
that contains its bases.
 Cylindric section: Section of a straight cylinder,
whose superior base is cut by an oblique plane.
 Circle: Closed section included within the
circumference.
 Circumference: Closed curved formed by the
collection of dots with equal distance from the
center.
 Radius: Distance from the center of the
circumference and any other dot in the border.
 Pi ( π ): Constant equal to the number of times




that the diameter fits into the circumference.
Geometrical object: Element that occupies a
volume in space.
Face: The region of a plane that limits a
geometrical object.
Axis: Line used as a reference to generate a
surface or geometrical object.
Generatrix: Line segment generated by the
constant movement of a flat figure.


Area: Surface or limited segment of a plane.
Volume: Tridimensional space occupied by an
object.
NAME: _______________________________________________________
GRADE: ______________________________________________________
Introduction
Geometrical objects with cylindrical shapes
Geometrical objects are found in our surroundings; they are part of
everyday life and have different shapes. (Some have special shapes,
with specific functions).
We can find regular and irregular solid geometrical objects. Regular
geometrical objects share the characteristic of having equal edges, faces
and angles. Irregular geometrical objects don’t have equal edges or
faces.
When talking about solid geometrical objects with the least amount of
faces, we can find the tetrahedron; however, a very particular irregular
geometrical object is the cylinder: It has few faces and a limitless
number of uses due to its curved face.
In present day, it is possible to find cylinders almost everywhere: simple
beverage containers (such as soda drinks), glasses, trash bins, complex
pinion and piston systems, and car tires. Its uses provide a wide range of
research possibilities.
For example, do you know how to calculate how much liquid is it possible
to introduce in a soda can? The amount of water you need to fill a glass?
Can you recognize the materials used to build a trash bin or a tire?
Think about these questions and write down some of your ideas.

___________________________________________________________

___________________________________________________________
Objectives
Develop processes to solve problems related with the shape, surface
area and cylinder volume.



Recognize cylindrical shapes starting from the elements that
compose them.
Characterize objects with cylindrical shapes based on the
surface area.
Identify objects with cylindrical shape with the information of
the volume.
Activity One
Cylinders
In order to define a cylinder, one must consider multiple aspects. Let’s
see some definitions:
Cylinders are geometric objects formed by two flat circular faces that
serve as base and top, and one lateral curved surface.
A more technical definition indicates that a cylinder is a cylindric surface
generated from a generatrix line that spins around another parallel line,
called axis.
The elements of a cylinder are:

Bases: The bases of a cylinder are those flat round surfaces, equal
and parallel that are generated when spinning the AB and DC
sides. These are perpendicular to the axis. AB and DC are the
radius of the cylinder.

Axis: AD segment. The axis of a cylinder is the fixed side around
which the rectangle spins.

Height: The perpendicular distance between the bases. It is also

known as the AD segment (in right cylinders) because it is equal to
the rotation axis.

Generatrix: the BC side that creates or generates the cylinder. It
is opposed and equal to the AD axis.
Types of cylinders
In geometry we can find 3 different types of cylinders:
1. Straight or right cylinder. It is called as such because its axis is
perpendicular to the base.
2. Oblique Cylinder. It is called like this because its axis is not
perpendicular to the base, it is oblique. It has an inclination degree
different from 90°. In this case, it is important to remember that
the height of said cylinder corresponds to the perpendicular
distance from the base up to the highest point of the cylinder.
3. Truncated Cylinder. Also known as Cylindric Section. It has
this name because its superior face has an oblique cut, thus
creating an ellipse as superior face, deforming the straight
cylinder. As a consequence, it loses the characteristic of having
parallel flat faces.
Learning Activity
Individual activity
Build each of the cylinders explained previously, based on the templates
or instructions you will find next.
Straight Cylinder
Cut and build according to the following image. (Pay attention to the
sleeves to paste it together)
Color in the chart as follows:
Bases: yellow
Lateral face: blue
Based on the blueprint and the construction product, find the value of:
Height: _________; Generatrix: _______; Radius of the bases: _____________
Oblique Cylinder. (Group Activity)
In this case, we will use one common material: coins. We will use ten
coins (or more, depending on the students´ choice) of the same value.
We will follow the instructions found next.
Remember! Before we begin, we must have a ruler and a pencil at hand.
Instructions:
1. Have 10 or more coins of the same value.
2. Place the coins on a flat, horizontal surface, one on top of the
other, creating a straight cylinder. (figure 6)
3. Use the ruler to create an oblique support regarding the horizontal
surface.
4. Next, in sequence, rest the coins over the ruler, as shown in the
image. (figure 7)
5. An oblique cylinder has been generated.
Find the values of:
Generatrix: ________;
Height: _________;
Radius of the bases: ________________.
Now change the angle of the ruler and generate a different cylinder.
Answer each of the following questions:
a. Which of the values found changed? What has to change for the
height to vary? ____________________________________________
______________________________________________________________________
________________________________________________________
b. Is the value of the generatrix equal to the height? In which cases
are they equal? ________________________________________
_________________________________________________________________________
___________________________________________________________
c. What is the relation between the angle created with the ruler and
the value of the generatrix? Explain this
phenomenon.__________________
_________________________________________________________________________
___________________________________________________________
Cylindric section (individual activity)
Cut the figure and build the structure.
Once the object is created, follow these instructions.

Color the base blue.

Color the lateral section yellow.

Color the elliptic face green.
From the construction find (if possible) the value of:
Inferior height: ______
Superior height: ________
Radius of the base: _____________
Activity 2
Areas and volumes of cylinders
As determined previously, cylinders are formed by different elements,
which in a specific way are represented in flat figures. Let´s see the case
of the straight cylinder:
The cylinder is made of two circular bases, a curved surface that, when
extended, forms a rectangle. Each one of them, by definition, has its own
area. Additionally, the following relationship can be established:
Surface area of a straight cylinder
Now, let´s see how we can calculate the surface area of a straight
cylinder:
First, it’s necessary to look at the areas of the round faces. For this,
consider the following formula:
A=π r 2
As there are two faces, then we have:
2
2π r
The lateral area of a rectangle can be calculated with the product of the
base multiplied by the height. The length of the base is 2 πr and the
height is determined by the value of
h
in the cylinder. Therefore, we
have:
A=¿
2 πr ×h
As a result, adding the areas will result in the total area of the cylinder:
A t =2 πr ×h+ ¿
2 π r2
Volume of a straight cylinder
The volume of a solid object is directly related with its capacity, keeping
in mind the 3 dimensions. In objects that are formed by figures with
parallel faces, the method is standardized. The volume can be found
multiplying the area of the base times the height of the object and a
straight cylinder is not excluded from the norm; therefore, its volume
can be found the area of the base.
A=π r
As a result, we have:
2
, times the height
V =π r 2 × h .
h .
Problem situation:
A soda bottling company has received a special order: 20 sodas in can
containers. The owner of the company wants to calculate the amount of
material that he needs to produce each can, and the amount of liquid
available to bottle.
The measures of the cans are: base radius 4cm and height 10cm.
Situation analysis:
2
First, we calculate the volume of each can: V =π r × h .
V =π ∙(4 cm)2 ×10 cm .
V ≈503 cm
3
.
Next, we look for the surface area:

To find the area of the bases we have:
A=π r 2
A=π ∙ ( 4 m )
2
A=50,265 cm 2 ≈ 50 cm2
As we have 2 bases, we multiply by 2.
Area of the bases:

2
A ≈ 100 cm
For the area of the rectangular face, we use the general formula:
We use the general formula:
A=¿
A=2∙ π ∙( 4 cm)∙(10 cm)
A=251,38 cm2 ≈ 251 cm2
2 πr ×h
Adding the areas, we find as a total:
2
A t =100 cm + 251cm
2
A t =351 cm2
Conclusion: To produce each can, we need a minimum of
351 cm2
of
can sheet, without considering the residue or sealing tabs; the volume or
3
capacity of each can is 503 cm
Volume of a cylindric section
Solve the following situations: In each case, give a stimated answer.

How many
cm 3
of water can be stored in a container that has a
10cm radius and 25cm height?

It is needed to cover with colored paper the curved surface of a
cylinder
with
the
following
dimensions:
r=3 cm, and height h=15 cm . How much material is required?
Reinforcement activity (Work in pairs)
Build two equal cylindric sections with the following directions:

Radius of the inferior base: 2cm

Height 1: 4cm

Height 2: 6cm
Place one on top of the other trying to generate one straight cylinder
and answer the following questions:
What is the height of the new cylinder?
__________________________________________________________________
Based on the dimensions of the new cylinder, is it possible to find the
volume of each cylinder section?
__________________________________________________________________
How could they be related to each other?
__________________________________________________________________
Find the volume of the cylinder section and the volume of the new
cylinder and compare the answers. Is there any relation between the
cylinders?
__________________________________________________________________
Leisure Activity (Fill in the blanks game)
Use the following words to complete the sentences: “straight”, “oblique”,
“cylindric section”, “heights”, and “base and height”.
a. The volume of oblique and straight cylinders can be found through the
product of the area of __________________.
b. Cylindric sections have two different ___________.
c. Round faces that are parallel are found in ________ and _______
cylinders.
d. _________________ are formed by cutting an oblique plane into a
straight cylinder.
Abstract
The following flow chart allows us to make a review of the topics covered
in the guide.
Homework
1. In groups of 2 or 3 students identify in your everyday surroundings
round geometrical objects, for each type of cylinders. Draw them
and label each one of its parts.
Share with your classmates the elements found and point out
the most common ones.
2. Mathematical challenge
Suggest a development strategy and share it in class.
The tires of a car are straight cylinders supported on a curved surface. Is
it possible to know the distance a car has traveled based on an
established number of revolutions of one tire? Suggest a strategy and
solve the following problem.

A car is travelling along a road; it is known that its tires have spun
200 times. If the radius of the tire is 30cm, what distance has the
car travelled?
o If it spun 500 times, what distance did the car travel?
o If it spun n times, what distance did the car travel?
3. Get a glass, a barrel, a soda can or any other object that has a
cylinder shape.
 Obtain its measurements and calculate its volume.
 Check that its capacity in cm3 is the same as indicated in the
label. If the cylinder does not have a capacity label, fill it with
water and verify the amount using a measured container such as a
blender cup to calculate the amount of water used to fill it.
 Remember that 1 liter equals to 1000 milliliters and 1 milliliter is
1cm3. (Keep in mind that the amounts used are estimated and the
results may have a small error percentage).
Evaluation
Based on the different definitions and types of cylinders, answer each
one of the following questions.
1. In ____________ cylinders, the length of the generatrix is equal to
the length of its height.
2. In oblique cylinders, the length of the ___________ will always be
greater than its height.
3. A ___________ is the flat figure that, when spinning one of its sides
as an axis, generates the cylinder.
Short answer questions:
Based on the following image, label the name of each element of a
straight cylinder.
4. AD:__________________
5. CB__________________
6. DC__________________
Problem Zone:
7. A wooden cylinder gets a cut like the one in the following image:
The volume of the new segment is needed; knowing that the radius is 2
cm, choose the closest answer.
a.
b.
c.
d.
100,6 cm3
50,3 cm3
25,1 cm3
12,6 cm3
8. It is common to find cylinders in kitchen paper towels. John took in the
task of investigating how much carton is needed for their construction.
For this, he measured the radius of the circumference and the length of
the roll, obtaining the following results: r=2,5 cm y h=20 cm
The estimated amount of carton that is used for each roll is:
a.
b.
c.
d.
314,2 cm2
618,3 cm2
76,6 cm2
157,1 cm2
Bibliography
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
Stewart, J., Redlin L. & Watson, S. (2012) Precálculo, Matemáticas
para el cálculo, sexta edición. Gengage Learning. México.
Boyer, C. (2003). Historia de la Matemática. Editorial Alianza:
Madrid.
González, G. () Matemáticas Opcion A. 4° ESO. Editex. España.
PlanCeibal. Cilindros. Inquiry date March 10th, 2016. URL:
http://www.ceibal.edu.uy/UserFiles/P0001/ODEA/ORIGINAL/110926_
cilindros.elp/index.html.