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Total surface area and volume of the pyramid

Source: licencja: CC 0.

Pole powierzchni całkowitej i objętość ostrosłupa

Learning objectives

You will learn to sketch the pyramid nets.

You will learn to calculate the total surface area and the volume of pyramids.

Learning effect

  • You will sketch pyramid nets.

  • You will calculate the total surface area and volume of pyramids.

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nagranie abstraktu

Review your knowledge of pyramids. Complete the sentences:

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

  2. The pyramidpyramid pyramid, whose side edges are equal in length, is called ......

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

  4. The pyramidpyramid pyramid, which has a n‑th angle in the base, has ...... edges.

  5. A regular tetrahedron is a ...... pyramid.

The aim of the lesson is to review and use the formulas to calculate the total surface area and volume of the pyramidpyramid pyramid.

Task 1
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nagranie abstraktu

Analyze the material presented in the applet. Write the appropriate formulas.

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Aplet geogebra: Pole powierzchni całkowitej i objętość ostrosłupa. Poniżej znajduje się galeria będąca wersją alternatywną dla aplikacji.
Source: GroMar, licencja: CC BY 3.0.
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nagranie abstraktu

Formulas for calculating the surface area and volume of the pyramidpyramid pyramid.

The volume V of the pyramidpyramid pyramid with the base area Pp and the height H is given by:

V=13·Pp·H

The total surface area of the pyramidtotal surface area of the pyramidtotal surface area of the pyramid Pc with the base area Pp and the area of the lateral surface Pb is given by the formula:

Pc=Pp+Pb
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nagranie abstraktu

Use the presented formulas in the word problems below.

Task 2
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nagranie abstraktu

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

Task 3
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nagranie abstraktu

The lateral area of the triangular pyramidpyramid pyramid is equal to 1363 cmIndeks górny 2 and the base area is 643 cm.

Task 4
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nagranie abstraktu

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

Task 5
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nagranie abstraktu

Calculate the volume of a triangular pyramidpyramid pyramid 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 6
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nagranie abstraktu

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

Task 7
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nagranie abstraktu

The perfume bottle has the shape of a pyramidpyramid pyramid whose base area is equal to 4 cmIndeks górny 2 and the height is equal to 9 cm. Marta has already used 7 cmIndeks górny 3 of perfume. Prove that the bottle contains less than 5,5 cmIndeks górny 3 of perfume.

Task 8

An extra task:

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nagranie abstraktu

Scouts built 8 sand castles on the beach. Each sand castle had the shape of a regular rectangular pyramidpyramid pyramid 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?

On completion of all the tasks do the revision exercises.

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nagranie abstraktu
  • The total surface area of the pyramidtotal surface area of the pyramidtotal surface area of the pyramid is equal to the sum of the base area and the area of its faces.

  • The pyramid volumepyramid volumepyramid volume equals to one third of the product of the base area and the height of the pyramidpyramid pyramid.

Exercises

Exercise 1
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Wersja alternatywna ćwiczenia: Determine which sentences are true. Możliwe odpowiedzi: 1. The volume of a regular tetrahedron with an edge length of 6 cm is equal to 182 cm3., 2. Each face of the regular quadrilateral pyramid has an area of 8. Its volume is equal to 2414., 3. The height of a regular triangular pyramid is six times longer than the edge of the base, and the volume of this pyramid is equal to 43. The edge length of the pyramid's base is 2., 4. The volume of the regular hexagonal pyramid whose base edge is 8 and the lateral edge is 10 is 1983.
zadanie
Source: GroMar, licencja: CC BY 3.0.
Exercise 2

Janek wants to make a cardboard box in the shape of a square pyramid, in which the edge of the base is 5 cm long and the height of the face is equal to 9 cm. Is a cardboard sheet in the shape of a square with a side length of 13 cm sufficient to make a box?

Exercise 3

A regular triangular pyramid is given. Its base edge is 5 cm long and the side edge length is 8 cm. Describe in English how you calculate the volume of this pyramid.

Exercise 4
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Wersja alternatywna ćwiczenia: Indicate which pairs of expressions or words are translated correctly. Możliwe odpowiedzi: 1. ostrosłup - pyramid , 2. ostrosłup prawidłowy - regular pyramid, 3. objętość ostrosłupa - pyramid volume, 4. pole powierzchni całkowitej ostrosłupa - total surface area of the pyramid, 5. krawędź podstawy - lateral edge, 6. krawędź boczna - base edge
zadanie
Source: GroMar, licencja: CC BY 3.0.
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Interaktywna gra, polegająca na łączeniu wyrazów w pary w ciągu jednej minuty. Czas zaczyna upływać wraz z rozpoczęciem gry. Jeden ruch to odkrywanie najpierw jednej potem drugiej karty z wyrazem. Każdy wyraz jest odczytywany. Kolejny ruch to odkrywanie trzeciej i czwartej karty. W ten sposób odsłuchasz wszystkie wyrazy. Nawigacja z poziomu klawiatury za pomocą strzałek, odsłuchiwanie wyrazów enterem lub spacją. Znajdź wszystkie pary wyrazów.
Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

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

kąt nachylenia krawędzi bocznej do podstawy ostrosłupa

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wymowa w języku angielskim: angle of inclination of the lateral edge to the base
angle of inclination of the lateral face to the base
angle of inclination of the lateral face to the base

kąt nachylenia ściany bocznej do podstawy ostrosłupa

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wymowa w języku angielskim: angle of inclination of the lateral face to the base
base edge
base edge

krawędź podstawy

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wymowa w języku angielskim: base edge
lateral edge
lateral edge

krawędź boczna

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wymowa w języku angielskim: lateral edge
lateral face height
lateral face height

wysokość ściany bocznej

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wymowa w języku angielskim: lateral face height
pyramid
pyramid

ostrosłup

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wymowa w języku angielskim: pyramid
pyramid height
pyramid height

wysokość ostrosłupa

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wymowa w języku angielskim: pyramid height
pyramid volume
pyramid volume

objętość ostrosłupa

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wymowa w języku angielskim: pyramid volume
regular pyramid
regular pyramid

ostrosłup prawidłowy

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wymowa w języku angielskim: regular pyramid
total surface area of the pyramid
total surface area of the pyramid

pole powierzchni całkowitej ostrosłupa

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wymowa w języku angielskim: total surface area of the pyramid

Keywords

pyramidpyramid pyramid

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