Topicmee38163692006551_1528449000663_0Topic

A coneconecone. Surface area of the conesurface area of the coneSurface area of the cone

Levelmee38163692006551_1528449084556_0Level

Third

Core curriculummee38163692006551_1528449076687_0Core curriculum

X. Solid geometry (Stereometry). The student:

6) calculates the volume and surface area of prisms, pyramids, cylinders, cones, balls, also using trigonometry and the theorems learned.

Timingmee38163692006551_1528449068082_0Timing

45 minutes

General objectivemee38163692006551_1528449523725_0General objective

Applying mathematical objects and manipulating them, interpreting mathematical concepts.

Specific objectivesmee38163692006551_1528449552113_0Specific objectives

1. Construction of the coneconecone.

2. Calculation of the surface area of the conesurface area of the conesurface area of the cone.

3. Communicating in English, developing competences in mathematics, science, technology and IT, developing learning skills.

Learning outcomesmee38163692006551_1528450430307_0Learning outcomes

The student:

- identifies the elements of the coneconecone.

- calculates the surface area of the conesurface area of the conesurface area of the cone.

Methodsmee38163692006551_1528449534267_0Methods

1. Discussion.

2. Mind map.

Forms of workmee38163692006551_1528449514617_0Forms of work

1. Individual work.

2. Whole class.

Lesson stages

Introductionmee38163692006551_1528450127855_0Introduction

The teacher informs students that during the lesson they will learn about the coneconecone elements and that they will learn to calculate its surface area.

Proceduremee38163692006551_1528446435040_0Procedure

Discussion - What figure will we get by rotating a right‑angled triangle around a line containing one of the legs?

Task
Students work individually using computers. Their task is to observe and identify the most important elements of the coneconecone construction.

[Geogebra applet]

Group work.

Students create mind maps, on which they show information about the coneconecone (also the one learnt in primary school). They mark the construction elements of the coneconecone and its sections on the drawing. The results of their work are presented to other groups.

From paper students cut out half‑circle. They mark its centre and radius. The section is rolled so that the rays marked by the intersection overlap. In this way they receive a lateral surface of the coneconecone. Discussion - is it always true that the lateral surface of the coneconecone has to be a circular sector of the diskcircular sector of the diskcircular sector of the disk? What shape does the base of the conebase of the conebase of the cone have? What elements is the cone netcone netcone net made of?

Conclusions:

[Illustration 1]

- The lateral surface of the cone, when unfolded on the plane, is a circular sector of the diskmee38163692006551_1527752256679_0The lateral surface of the cone, when unfolded on the plane, is a circular sector of the disk.

- The cone netcone netcone net consists of a disk, which is the base of the conebase of the conebase of the cone and the sector of the disk, which is the lateral surface.

- The coneconecone forming is equal to the radius of the sector of the disk, which is the lateral surface of the cone.

- The circumference of the cone base is equal to the length of the arc defined by the circular sectormee38163692006551_1527752263647_0The circumference of the cone base is equal to the length of the arc defined by the circular sector.

The surface area of the conesurface area of the conesurface area of the cone is equal to the area of the net of this coneconecone.

The surface area $P$ of the coneconecone with the base radius $r$ and forming $l$ is equal:

Students use acquired knowledge to solve the word problems.

Task
The cone‑shaped length of $4\sqrt{3}$ is inclined to the base at an angle of $30°$. Calculate the lateral surface area of the conelateral surface area of the conelateral surface area of the cone.

Task
The axial section of the coneconecone is an equilateral triangle with a side length of 4 cm. Calculate the total surface area of the conesurface area of the conesurface area of the cone.

Task
The lateral surface area of the conelateral surface area of the conelateral surface area of the cone is a sector of the disk with an angle 144° and a radius of 12 cm. Calculate the total surface area of the conesurface area of the conesurface area of the cone.

Task
A right‑angled triangle with a length of 5 cm and 12 cm has been rotated first around the shorter side, then around the longer side. Which cone has a larger surface area?

An extra task:
The opening angle of the coneopening angle of the coneopening angle of the cone has a measure of 120°, and the generatrix has a length of 8 cm. Calculate the base area of the cone.

Lesson summarymee38163692006551_1528450119332_0Lesson summary

Students do additional exercises.

They summarize the lesson and formulate the conclusions to remember.

- The lateral surface of the coneconecone, when unfolded on the plane, is a circular sector of the diskcircular sector of the diskcircular sector of the disk.

- The cone netcone netcone net consists of a disk, which is the base of the conebase of the conebase of the cone and the segment of the circle, which is the lateral surface.

- The coneconecone forming is equal to the radius of the segment of the circle, which is the lateral surface of the cone.

- The circumference of the coneconecone base is equal to the length of the arc defined by the circle segment.

The surface area of the conesurface area of the conesurface area of the cone is equal to the area of the net of this coneconecone.

The surface area $P$ of the coneconecone with the base radius $r$ and forming $l$ is equal:

Selected words and expressions used in the lesson plan

base of the conebase of the conebase of the cone

circular sector of the diskcircular sector of the diskcircular sector of the disk

coneconecone

cone netcone netcone net

lateral surface area of the conelateral surface area of the conelateral surface area of the cone

opening angle of the coneopening angle of the coneopening angle of the cone

surface area of the conesurface area of the conesurface area of the cone