Topicmcdebbc14d13cef2f_1528449000663_0Topic

The surface areasurface areasurface area and the volume of the cylindervolume of the cylindervolume of the cylinder

Levelmcdebbc14d13cef2f_1528449084556_0Level

Third

Core curriculummcdebbc14d13cef2f_1528449076687_0Core curriculum

X. Solid geometry. The student:

4. recognizes an angle between segments and an angle between segments and planes in cylinders and cones (e.g. a line inclination angle, an angle between the slant height and the base of the cone) and works out the sizes of these angles;

6. calculates the volumevolumevolume and the surface areasurface areasurface area of a prism, pyramid, cylinder, cone, sphere using trigonometry and theorems;

7. uses the relations between the volumes of similar solids.

Timingmcdebbc14d13cef2f_1528449068082_0Timing

45 minutes

General objectivemcdebbc14d13cef2f_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesmcdebbc14d13cef2f_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Calculating the surface areasurface areasurface area and volumevolumevolume of a cylinder using trigonometry.

Learning outcomesmcdebbc14d13cef2f_1528450430307_0Learning outcomes

The student:

- calculates the surface areasurface areasurface area and volume of the cylindervolume of the cylindervolume of the cylinder using trigonometry.

Methodsmcdebbc14d13cef2f_1528449534267_0Methods

1. Mind map.

2. Situation analysis.

Forms of workmcdebbc14d13cef2f_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmcdebbc14d13cef2f_1528450127855_0Introduction

The teacher informs the student that the aim of the class is calculating the surface areas and volumes of cylinders.

Working in groups students make mind maps of all the information about cylinders that they have already acquired. Then, they make presentations of their posters, paying special attention to the formulas for calculating the surfacesurface areasurface area and the volume of the cylindervolume of the cylindervolume of the cylinder.

Proceduremcdebbc14d13cef2f_1528446435040_0Procedure

The formula for the surface area of the cylinder:

$P_c=2\cdot P_p+2\cdot \pi \cdot r\cdot H$

r – the base radius of the cylinder,
H – the height of the cylinder.

The formula for the volume of the cylinder:

$V=\pi \cdot r^2\cdot H$mcdebbc14d13cef2f_1527752263647_0The formula for the surface area of the cylinder:

$P_c=2\cdot P_p+2\cdot \pi \cdot r\cdot H$

r – the base radius of the cylinder,
H – the height of the cylinder.

The formula for the volume of the cylinder:

$V=\pi \cdot r^2\cdot H$

Students work in groups analysing the material presented in the applet.

Task
Analyse carefully the material presented In the applet. Pay attention to the subsequent stages of solving the problem.

[Geogebra applet]

Working individually, students use the formulas to solve the tasks.

Task
A rectangle sized 10 cm by 15 cm was first rotated around its longer side and then it was rotated around its shorter side. Do both the cylinders have the same volumevolumevolume and total surface areasurface areasurface area? Justify your answer.
Answer: No.

Task
The axial section of a cylinder is a rectangle whose diagonal is 15 cm and it makes an angle $\alpha$ with the height. Calculate the height of the cylinderheight of the cylinderheight of the cylinder and its base radius when you know that $tg\alpha =3$.
Answer: H = $\frac{3}{2}\sqrt{10}$ cm, r = $\frac{3}{4}\sqrt{10}$ cm.

Task
The curved surface area of the cylinder unfolded on a plane is a rectangle. The longer side of this rectangle is the height of the cylinderheight of the cylinderheight of the cylinder. The diagonal of the rectangle is 18 cm and it makes an angle of 30° with the longer side of the rectangle. Calculate the volume of the cylindervolume of the cylindervolume of the cylinder.
Answer: V = $\frac{729\sqrt{3}}{4·\pi }$.

Task
Two cylinders are similar on the scale of k = 3. The sum of the volumes of both cylinders equals $140·\pi$. Calculate the volumevolumevolume of each of the cylinders. Justify your answer.
Answer: $V_1=5·\pi$, $V_2=135·\pi$.

Task
The sum of the surface areas of both bases of a cylinder equals its curved surface areasurface areasurface area and measures $72·\pi$ cmIndeks górny 2². Calculate the height of the cylinderheight of the cylinderheight of the cylinder.
Answer: H = 6 cm.

Task
The inside diameter of each of the pipes that make a water supply system measures 3 inches (1 inch = 2,54 cm) In the water supply system there are about 510 l of water. Calculate what is the approximate length of the water supply system.
Answer: 120,2 km.

Having solved all the tasks the students sort the result numbers from the largest to the smallest.

An extra task
The mass of a cylinder shaped rotor, whose diameter is 6 cm and the height 10 cm, equals 1,4 kg. Calculate the mass of 1 cmIndeks górny 3³ the alloy which was used to make the rotor.
Answer: 0,005 kg.

Lesson summarymcdebbc14d13cef2f_1528450119332_0Lesson summary

Students do the consolidation tasks. They cooperate to make the formulas to be remembered.

- The total surface area of the cylinder equals the sum of the surface areas of its bases and the curved surface area.
- The volume of the cylinder equals the product of the surface area of its base and its height.mcdebbc14d13cef2f_1527752256679_0- The total surface area of the cylinder equals the sum of the surface areas of its bases and the curved surface area.
- The volume of the cylinder equals the product of the surface area of its base and its height.

Selected words and expressions used in the lesson plan

axial section of the cylinderaxial section of the cylinderaxial section of the cylinder

base of the cylinderbase of the cylinderbase of the cylinder

curves surface area of the cylindercurves surface area of the cylindercurves surface area of the cylinder

cylindercylindercylinder

diameter of the axial section of the cylinderdiameter of the axial section of the cylinderdiameter of the axial section of the cylinder

height of the cylinderheight of the cylinderheight of the cylinder

surface areasurface areasurface area

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

volumevolumevolume

volume of the cylindervolume of the cylindervolume of the cylinder