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Pyrámids

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Pyrámids

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About this activity

In this activity you will find the properties of the pyramids and how they vary according to their class.

Created by

karen dayana campuzano campo
karen dayana campuzano campo
Colombia

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Pyrámids
 

PyrámidsOnline version

In this activity you will find the properties of the pyramids and how they vary according to their class.

by karen dayana campuzano campo
1

Pyramids

The Pyramids

The pyramids are geometric solid bodies that are composed of a base that is any polygon, and their lateral faces are triangles that join each other by a common vertex.


2

Part pyramid

Parts

Base: this is the name given to the face where the pyramid rests.

Faces: are the geometric figures that form the walls of the pyramid.

Edges: segments of the line that join two faces.

Vertices: these are the points of union of the edges.

Cusp: it is the highest point of the pyramid.

Height: distance from the center of the base to the top of the pyramid.

Apothema of the pyramid (ap): it is known as the distance that exists between the vertex and one side of the base, and as the height of the lateral faces of the pyramid. It can only be observed in the regular pyramids.

Apothema of the base (apb): is the distance that exists between one side of the base and the center of it. It is important to know that it can only be observed in regular pyramids

3

Types of pyramids

Types: The pyramids are named according to their base.

 

Pentagonal: In geometry, a pentagonal pyramid is a pyramid based on a pentagon hence the name pentagonal pyramid, that is, its base has five sides on which five triangular faces are governed that are at one point, the cusp. Like any pyramid, it is self-dual. This polyhedron has 6 faces, 10 edges and 6 vertices.

4

hectagonal pyramids

Hexagonal pyramid

A hexagonal pyramid is a geometric figure in three dimensions, whose base is a hexagon (six-sided polygon) and which also has six triangular faces, which meet at a certain height of the base, at a point called apex or vertex, that is, it has 7 faces, 12 edges and 7 vertices

5

Rectangular pyramids

Rectangular pyramids

Rectangular pyramids are three-dimensional figures formed by a base and side faces. The base has a rectangular shape and the side faces are triangles. Rectangular pyramids have 5 faces, 8 edges and 5 vertices. These figures are not regular, as their base has sides of different lengths. However, the opposite faces are the same.
6

Triangular pyramid

Triangular pyramid

A triangular pyramid is a three-dimensional figure, in which all its faces are triangles. These pyramids are characterized by having a triangular base and three lateral triangular faces. Triangular pyramids have 4 faces, 6 edges and 4 vertices.

7

Clacification of pyramids

8

Formulas for the area and volume of a pyramid

formulas of the area and volume of a pyramid

Formulas for the area:

A = A base + ∑A Lateral face

Formula for volume:

V = L∙ ap base∙ h


9

example

The surface area A of the hexagonal pyramid, either regular or irregular, is calculated by adding the areas of the lateral faces and the area of the hexagonal base:

A = Abase + ∑Lateral face

In the formula, the symbol "∑" represents a sum, to indicate in summary form the sum of the six areas of the lateral faces.

For the regular hexagonal pyramid there is a formula to find the area:

A = 3L∙ (apbase + appirámide)

Where:

• L is an edge of the base (the side of the hexagon).

• apbase is the apothema of the base

• appiramid is the apothema of the pyramid.

If the pyramid is not regular, either because the base is not a regular hexagon or because the pyramid is oblique, it is necessary to calculate the areas of each separately and then add.

The regular hexagonal pyramid also has a formula for volume:

V = L∙ apbase∙ h

Here "h" represents the height of the pyramid.

And if the hexagonal pyramid is not regular, there is a general formula, applicable to all pyramids, to calculate its volume:

V = 1/3∙ Abase ∙ h


10

Numerical example

Numerical example

For the regular hexagonal pyramid whose dimensions are:

Apothema of the base: 4 cm

Base edge length: 7 cm

Apothema of the pyramid: 15 cm

Height: 10 cm

Calculate the following:

a) Area of the hexagonal base.

b) Surface area of the pyramid.

(c) Volume

Solution to

The area of a regular hexagon is:

A = 1/2 (Perimeter × apothema) = 1/2 (6L× apbase)

A = 3L∙ apbase = 3×7cm × 4cm = 84cm2

Solution b

A = 3L∙ (apbase + appirámide) = 3L∙ apbase + 3L∙ appirámide = 84cm2 + (3×7cm×15cm) = 399 cm2.

Solution c

The volume can be found by the general formula:

V = 1/3∙ Abase ∙ h = 1/3∙ 84cm2 ∙10cm = 280 cm3
 
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