Topicm4501d6b32d2ac6b4_1528449000663_0Topic

The surface area and the volume of the conevolume of the conevolume of the cone

Levelm4501d6b32d2ac6b4_1528449084556_0Level

Third

Core curriculumm4501d6b32d2ac6b4_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 conebase of the conebase of the cone) and works out the sizes of these angles;

6) calculates the volume and the surface area of a prism, pyramid, cylinder, coneconecone, sphere using trigonometry and theorems.

Timingm4501d6b32d2ac6b4_1528449068082_0Timing

45 minutes

General objectivem4501d6b32d2ac6b4_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 objectivesm4501d6b32d2ac6b4_1528449552113_0Specific objectives

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

2. Calculating the surface area and volume of a coneconecone using appropriate formulae.

3. Practical application in everyday life situations.

Learning outcomesm4501d6b32d2ac6b4_1528450430307_0Learning outcomes

The student:

- calculates the surface area and volume of a coneconecone using appropriate formulae,

- uses the surface area and the volume of the conevolume of the conevolume of the cone formulae in everyday life situations.

Methodsm4501d6b32d2ac6b4_1528449534267_0Methods

1. Incomplete sentences.

2. Situation analysis.

Forms of workm4501d6b32d2ac6b4_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm4501d6b32d2ac6b4_1528450127855_0Introduction

The teacher informs the students that the aim of the class is developing the ability to calculate the surface area and the volume of cones.

Procedurem4501d6b32d2ac6b4_1528446435040_0Procedure

Students use the incomplete sentences technique to get the information about the coneconecone in order.

- The coneconecone is a solid formed by making a right‑angled triangle revolve around …

- The apex angle of the coneapex angle of the coneapex angle of the cone is the angle …

- The slant height of the coneslant height of the coneslant height of the cone is a line segment between …

- The net of the cone consists of …

- The curved surface of the cone is …

The teaches assesses the students’ work and explains doubts.

Task
Working in groups, students analyse an interactive diagram presenting the formula for calculating the total surface area of a coneconecone. They take notes of appropriate relations.

[Interactive ilustration 1]

The formula for calculating the total surface area of the cone:

$P_c=\pi ·r·l+\pi ·r^2$

PIndeks dolny c_c - the total surface area of the cone,
r – the base radius of the cone,
l – the slant height of the cone.m4501d6b32d2ac6b4_1527752263647_0The formula for calculating the total surface area of the cone:

$P_c=\pi ·r·l+\pi ·r^2$

PIndeks dolny c_c - the total surface area of the cone,
r – the base radius of the cone,
l – the slant height of the cone.

Students use the information to solve the tasks.

Task
The slant height of a coneconecone unfolded on a plane is a circle sector whose radius is 15 cm and the central angle is 130°. Calculate the base radius of the conebase radius of the conebase radius of the cone.
Answer: $5\frac{5}{12}$ cm.

Students work in groups. The teacher gives a cylinder‑shaped container and several identical cone‑shaped containers to each group of the students. The cones and the cylinder have identical heights and bases. The students’ task is to find relations between the volumes of both containers. In order to do so they fill the coneconecone with water and check how many times the volume of a cone is smaller than the volume of a cylinder. The students formulate hypotheses and conclusions.

Conclusion:

The volume of the cone is three times smaller than the volume of the cylinder. Both solids have identical height and identical base radius.m4501d6b32d2ac6b4_1527752256679_0The volume of the cone is three times smaller than the volume of the cylinder. Both solids have identical height and identical base radius.

Task
Analyse carefully the interactive diagram. Write down the formula.

[Interactive ilustration 2]

The formula for calculating the volume of the conevolume of the conevolume of the cone:

V – the volume of the conevolume of the conevolume of the cone,

r – the base radius of the conebase radius of the conebase radius of the cone,

H – the height of the coneheight of the coneheight of the cone.

Students solve the problems individually using the acquired information.

Task
The circle sector with a right‑angled central angle and the radius 8 cm long makes the curved surface. Calculate the volume of this coneconecone and the sine of the inclination angle between the vertical height and the slant height of this cone.
Answer: $V=\frac{8·\mathrm{\pi }\sqrt{15}}{3}$ cmIndeks górny 3³, $\sin \alpha =\frac{1}{4}$.

Task
A cylinder and a coneconecone have identical heights and volumes. The base radius of the cylinder equals r. Calculate the base radius of the conebase radius of the conebase radius of the cone.
Answer: $r\sqrt{3}$.

Task
The right triangle with legs 5 cm and 7 cm was first revolved around the shorter and then the longer leg. Do both the obtained cones have identical volumes and total surface areas? Justify your answer.
Answer: No.

Task
A cone‑shaped tent has a radius of 1m. 7,5 mIndeks górny 2² of the material was used to make the tent without the floor. Calculate the height of the tent, assuming that $\mathrm{\pi }\approx 3$.
Answer: 2,3 m.

Task
How will the volume of the conevolume of the conevolume of the cone change if we make the height 3 times smaller and the radius 3 time larger?
Answer: The volume will be 3 times larger.

An extra task
A right angled triangle with legs measuring 8 cm and 6 cm was revolved around the hypotenuse. Calculate the volume and the total surface area of the obtained solid.
Answer: $P_c=67,2·\mathrm{\pi }$ cmIndeks górny 2², $V=76,8·\mathrm{\pi }$ cmIndeks górny 3³.

Lesson summarym4501d6b32d2ac6b4_1528450119332_0Lesson summary

Students do the consolidation tasks.

They work together to formulate conclusions and formulae to be remembered.

- The formula for calculating the total surface area of the cone:

$P_c=\pi ·r·l+\pi ·r^2$

- The volume of the cone is three times smaller than the volume of the cylinder. Both solids have identical height and identical base radius.
- The formula for calculating the volume of the cone:

$V=\frac{1}{3}·\mathrm{\pi }·{\mathrm{r}}^2·\mathrm{H}$

Selected words and expressions used in the lesson plan

apex angle of the coneapex angle of the coneapex angle of the cone

axial section of the coneaxial section of the coneaxial section of the cone

base of the conebase of the conebase of the cone

base radius of the conebase radius of the conebase radius of the cone

coneconecone

curved surface area of the conecurved surface area of the conecurved surface area of the cone

height of the coneheight of the coneheight of the cone

slant height of the coneslant height of the coneslant height of the cone

total surface area of the conetotal surface area of the conetotal surface area of the cone

volume of the conevolume of the conevolume of the cone