Calculating the surface area and the volume of the cylinder

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Obliczanie pola powierzchni i objętości walca

Learning objectives

You will develop the following competences: communication in English, mathematical, IT and basic scientific and technical competence, your learning skills.
You will get to know the method of calculating the surface area and volume of a cylinder using trigonometry.

Learning effect

You will learn how to calculate the surface area and volume of the cylinder using trigonometry.

Recollect information about the cylindercylindercylinder and the formula for calculating the surface area and volume of the cylindercylindercylinder. The aim of the class is developing the ability to calculate the surface areas and volumes of the cylindercylindercylinder.

Analyse the material shown in the multimedia presentation, paying special attention to the following stages of solving the task. Formulate your conclusion.

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In order to solve a task concerning calculating the surface areas and volumes of the cylinder, you need to do the following activities:

make a diagram of the solid,
find the given information and decide what information is needed to solve the task,
list the formulae appropriate for the task,
do the calculations,
formulate your answer.

Use the information to solve the tasks.

Exercise 1

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nagranie abstraktu

The diagonal of the axial section of a cylindercylindercylinder has the length of 26 cm and forms angle $α$ with the base of the cylinderbase of the cylinderbase of the cylinder so that $\cos α = \frac{5}{13}$ . Calculate the volume of the cylindercylindercylinder.

$600 π c m^{3}$

Exercise 2

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nagranie abstraktu

The curved surface of a cylindercylindercylinder equals $48 π d m^{2}$ and the perimeter of its base equals $8 π$ dm. Calculate the volume of the cylindercylindercylinder.

$96 π d m^{3}$

Exercise 3

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nagranie abstraktu

a) The radius of the base,
b) the altitude of the cylinderaltitude of the cylinderaltitude of the cylinder,
were increased three times. How many times has the volume of the cylindercylindercylinder increased?

Exercise 4

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nagranie abstraktu

Daniela wants to paint 10 identical fence post. The posts have the shape of a cylindercylindercylinder whose altitude is 2 m and the diameter of the base is 10 cm. How many cans of paint must Daniela buy if you know that one can of paint is enough to paint $5 m^{2}$ of the surface?

Exercise 5

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nagranie abstraktu

In a meadow there are 6 drainage ditches. Each ditch has a shape of a cylindercylindercylinder whose diameter is 30 cm and the length is 2 m. Calculate how many litres of water there are in the ditches. Give your calculation to the accuracy of 1 l.

Exercise 6

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nagranie abstraktu

An extra task:

A drawer has the length of 40 cm, the width 60 cm and the depth 15 cm. We put the maximum number of identical tins full of powdered chocolate into the drawer. Each tin has a shape of a cylindercylindercylinder with the altitude of 15 cm and the diameter of 10 cm. How big part of the drawer would be filled if the content of the tins was poured into the drawer?

Having finishes all the tasks, do the consolidation tasks. Formulate the conclusions to memorize.
In order to solve a task concerning calculating the surface areas and volumes of the cylinder, you need to do the following activities:

make a diagram of the solid,
find the given information and decide what information is needed to solve the task,
list the formulae appropriate for the task,
do the calculations,
formulate your answer.

Exercises

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Exercise 7

Determine which sentences are true. Możliwe odpowiedzi: 1. The volume of the cylinder whose base radius is

3 c m

and the altitude is

10 c m

equals

300 π

., 2. A square whose area is

324 c m^{2}

is rotated around its side. The total surface area of the cylinder formed in this way equals

1296 π c m^{2}

., 3. The cylinder whose base radius is

12 c m

was formed by a rotation of a rectangle whose diagonal equals

13 c m

. The volume of this cylinder equals

720 π c m^{3}

., 4. After unfolding the curved surface of a cylinder on a plane we get a square whose side is

4 π

. The volume of this cylinder equals

16 π

Determine which sentences are true.

The volume of the cylinder whose base radius is $3 c m$ and the altitude is $10 c m$ equals $300 π$ .
A square whose area is $324 c m^{2}$ is rotated around its side. The total surface area of the cylinder formed in this way equals $1296 π c m^{2}$ .
The cylinder whose base radius is $12 c m$ was formed by a rotation of a rectangle whose diagonal equals $13 c m$ . The volume of this cylinder equals $720 π c m^{3}$ .
After unfolding the curved surface of a cylinder on a plane we get a square whose side is $4 π$ . The volume of this cylinder equals $16 π$ .

zadanie

Source: GroMar, licencja: CC BY 3.0.

Exercise 8

The engine casing of the sewing machine has the shape of a hollow cylinder whose outside diameter is 26 cm, the inside diameter is 24 cm and the altitude is 30 cm. The casing is made of iron whose density is 720 $\frac{k g}{m^{2}}$ . Calculate the mass of the casing.

Exercise 9

The axial section of a cylinder is a square whose area is $400 c m^{2}$ . Describe in English how to calculate the sum of the areas of the two cylinder bases.

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Exercise 10

Indicate which pairs of expressions or words are translated correctly.

walec - cylinder
podstawa walca - base of the cylinder
wysokość walca - altitude of the cylinder
objętość walca - volume of the cylinder
pole powierzchni całkowitej walca - axial section of the cylinder
powierzchnia boczna walca - altitude of the cylinder

zadanie

Source: GroMar, licencja: CC BY 3.0.

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Match Polish terms with their English equivalents.

base of the cylinder
cylinder
podstawa walca
pole powierzchni całkowitej walca
wysokość walca
total surface area of the cylinder
volume of the cylinder
walec
altitude of the cylinder
objętość walca

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

altitude of the cylinder

wysokość walca

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wymowa w języku angielskim: altitude of the cylinder

axial section of the cylinder

przekrój osiowy walca

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wymowa w języku angielskim: axial section of the cylinder

base of the cylinder

podstawa walca

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wymowa w języku angielskim: base of the cylinder

curved surface area of the cylinder

powierzchnia boczna walca

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wymowa w języku angielskim: curved surface area of the cylinder

cylinder

walec

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wymowa w języku angielskim: cylinder

diameter of the axial section of the cylinder

przekątna przekroju osiowego walca

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wymowa w języku angielskim: diameter of the axial section of the cylinder

total surface area of the cylinder

pole powierzchni całkowitej walca

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: total surface area of the cylinder

volume of the cylinder

objętość walca

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: volume of the cylinder

Keywords

altitude of the cylinderaltitude of the cylinderaltitude of the cylinder

base of the cylinderbase of the cylinderbase of the cylinder

cylindercylindercylinder

total surface area of the cylindertotal surface area of the cylindertotal surface area of the cylinder

volume of the cylindervolume of the cylindervolume of the cylinder

Scenariusz

Exercises

Glossary

Keywords