Topicmca85b2e489707c76_1528449000663_0Topic

Total surface area and volume of the pyramidpyramidpyramid

Levelmca85b2e489707c76_1528449084556_0Level

Third

Core curriculummca85b2e489707c76_1528449076687_0Core curriculum

X. Solid geometry (Stereometry). Student:

3) Recognizes, in prisms and pyramids, angles between line segments (e.g. edges and edges or edges and diagonals); angles between faces; calculates the measures of these angles;

6) calculates the volume and surface area of prisms, pyramids, cylinder, cone, sphere, also using trigonometry and theorems learned;

7) uses dependencies between volumes of similar solids.

Timingmca85b2e489707c76_1528449068082_0Timing

45 minutes

General objectivemca85b2e489707c76_1528449523725_0General objective

Interpreting and manipulating information presented in the text, both mathematical and popular science, as well as in the form of graphs, diagrams, tables.

Specific objectivesmca85b2e489707c76_1528449552113_0Specific objectives

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

2. Sketching pyramidpyramidpyramid nets.

3. Calculation of the total surface area and volume of pyramids.

Learning outcomesmca85b2e489707c76_1528450430307_0Learning outcomes

The student:

- sketches the pyramidpyramidpyramid nets,

- calculates the total surface area and the volume of pyramids.

Methodsmca85b2e489707c76_1528449534267_0Methods

1. Incomplete sentences.

2. Situational analysis.

Forms of workmca85b2e489707c76_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmca85b2e489707c76_1528450127855_0Introduction

Students use incomplete sentences to review their knowledge of pyramids.

a) A polyhedron, whose base is any polygon, and whose faces are triangles with a common vertex, is commonly called ...

b) The pyramidpyramidpyramid, whose side edges are equal in length, is called ...

c) The pyramid, whose side edges are of equal length and in the base there is a regular polygon, is called ...

d) The pyramid, which has a n‑th angle in the base, has ... edges.

e) A regular tetrahedron is a … pyramidpyramidpyramid.

Proceduremca85b2e489707c76_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to review and use the formulas to calculate the surface area and volume of the pyramidpyramidpyramid.

Task
Students, working in groups, analyze the material presented in the applet. They write the appropriate formulas.

[Geogebra applet]

Formulas for calculating the surface area and volume of the pyramidpyramid pyramid.
The volume $V$ of the pyramidpyramid pyramid with the base area $P_p$ and the height $H$ is given by:

The total surface area of the pyramidtotal surface area of the pyramidtotal surface area of the pyramid $P_c$ with the base area $P_p$ and the area of the lateral surface $P_b$ is given by the formula:

Students use the formulas in word problems.

Task
Calculate the total surface area of the regular tetrahedron with the edge length of 12 cm.

Task
The area of the lateral triangular pyramid is equal to $136\sqrt{3}$ cmIndeks górny 2² and the base area is $64\sqrt{3}$ cmIndeks górny 2². Calculate the height of this pyramidpyramidpyramid.

Task
The total surface area of the square pyramid is equal to 20. The base area is four times smaller than the area of the lateral surface. Calculate the height of the pyramid and its volume.mca85b2e489707c76_1527752263647_0The total surface area of the square pyramid is equal to 20. The base area is four times smaller than the area of the lateral surface. Calculate the height of the pyramid and its volume.

Task
Calculate the volume of a triangular pyramidpyramidpyramid with a base edgebase edgebase edge length of 4 cm and an angle of inclination of the side edge to the plane of the base with a measure of 60°.

Task
In the right hexagonal pyramid, the longer diagonal of the base is equal to 6, and the angle between thelateral edgelateral edge lateral edgelateral edgelateral edge and the base is 60°. Calculate the volume of the solid. lateral edgelateral edge

Task
The perfume bottle has the shape of a pyramidpyramidpyramid whose base area is equal to 4 cmIndeks górny 2² and the height is equal to 9 cm. Marta has already used 7 cmIndeks górny 3³ of perfume. Prove that the bottle contains less than 5,5 cmIndeks górny 3³ of perfume.

An extra task
Scouts built 8 sand castles on the beach. Each sand castle had the shape of a regular rectangular pyramidpyramidpyramid with a height of 78 cm and the edge of the base with a length of 1 m. Would the sand used to build the castles be enough to fill a rectangular box shaped sandpit with dimensions of 180 cm ∙ 2 m ∙ 25 cm?

Lesson summarymca85b2e489707c76_1528450119332_0Lesson summary

Students do the revision exercises. Together, they formulate information to remember.

- The total surface area of the pyramid is equal to the sum of the base area and the area of its faces.
- The pyramid volume equals to one third of the product of the base area and the height of the pyramid.mca85b2e489707c76_1527752256679_0- The total surface area of the pyramid is equal to the sum of the base area and the area of its faces.
- The pyramid volume equals to one third of the product of the base area and the height of the pyramid.

Selected words and expressions used in the lesson plan

angle of inclination of the lateral edge to the baseangle of inclination of the lateral edge to the baseangle of inclination of the lateral edge to the base

angle of inclination of the lateral face to the baseangle of inclination of the lateral face to the baseangle of inclination of the lateral face to the base

base edgebase edgebase edge

lateral edgelateral edgelateral edge

lateral face heightlateral face heightlateral face height

pyramidpyramidpyramid

pyramid heightpyramid heightpyramid height

pyramid volumepyramid volumepyramid volume

regular pyramidregular pyramidregular pyramid

total surface area of the pyramidtotal surface area of the pyramidtotal surface area of the pyramid