unit 8. polyhedra (6)

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IES VIRREY MORCILLO. VILLARROBLEDO (ALBACETE)UNIT 8. POLYHEDRA

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UNIT 8.

POLYHEDRA




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INDEX

1. LET’S REMEMBER .2. POINT, LINES AND PLANES IN SPACE.2.1. POSITION OF TWO LINES IN SPACE.2.2. POSITION OF A LINE AND A PLANE IN SPACE.2.3. POSITION OF TWO PLANES IN SPACE.2.4. DIEDRAL ANGLE.

3. POLYHEDRA.3.1. ELEMENTS OF A POLYHEDRO.3.2. UNFOLD OF POLYHEDRAL.

4. REGULAR POLYHEDRAL.4.1. TETRAHEDRON.4.2. OCTAHEDRON.4.3. ICOSAHEDRON.4.4. HEXAHEDRON OR CUBE.4.5. DODECAHEDRON.

5. EULER’S FORMULA.6. PRISMS.

6.1. TYPES OF PRISMS.6.2. SURFACE AREA OF A RIGHT PRISM.

7. PARALLELEPIPEDS.7.1. TYPES OF PARALLELEPIPEDS.7.2. CUBOIDS.7.3. CUBES.7.4. DIAGONAL OF A CUBOID.

8. PYRAMIDS.8.1. PARTS OF A PYRAMID.8.2. TYPES OF PYRAMIDS.8.3. UNFOLD OF A PYRAMID.

8.4.

CALCULATION OF THE APOTHEM OF A PYRAMID.8.5. CALCULATION OF THE LATERAL EDGE OF A PYRAMID.8.6. SURFACE AREA OF A PYRAMID.

9. TRUNCATED PYRAMIDS.9.1. ELEMENTS OF A TRUNCATED PYRAMID.9.2. UNFOLD OF A TRUNCATED PYRAMID.9.3. CALCULATION OF THE APOTHEM OF A TRUNCATED

PYRAMID.9.4. SURFACE AREA OF A TRUNCATED PYRAMID.

10. EXERCISES AND PROBLEMS.

11. SELF EVALUATION.




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1. LET’S REMEMBER.

Polygons are 2-dimensional shapes. They are made of straight lines, and the

shape is “closed” (all the lines connect up).

Polygon (straight sides)

Not a polygon

(has a curve)Not a polygon

(open, not closed)

If all angles are equal and all sides are equal, then it is regular, otherwise it is

irregular.

Regular Irregular

A triangle is a polygon that it has three sides and three angles. The three

angles always add to 180°.

There are three special names given to triangles that tell how many sides (or

angles) are equal: Equilateral triangle, isosceles triangle and scalene triangle.

Equilateral Triangle

Three equal sides.

Three equal angles, always 60°.

Isosceles Triangle

Two equal sides.

Two equal angles.

Scalene Triangle

No equal sides.

No equal angles.




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Triangles can also have names that tell you what type of angle is inside:

Acute Triangle

All angles are less than 90°.

Right Triangle

Has a right angle (90°).

Obtuse Triangle

Has an angle more than 90°.

Quadrilateral just means “four sides” (quad means four, lateral means side).

So, quadrilateral is a polygon of four sides.

There are special types of quadrilateral:

There are special types of quadrilateral:

A parallelogram is a quadrilateral in which both pairs of opposite sides are

parallel. Squares, rectangles and rhombuses are parallelograms. Other parallelograms

are called non-parallelogram.

A rectangle is a parallelogram that has equal opposite sides and four right

angles.

A square is a parallelogram that has four right angles and four equal sides.




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A circle is a closed curved line whose points are all the same distance from a

fixed point called the center.

The circumference is the length around a circle, that is to say, it is the perimeter

of the circle. It is exactly Pi (the symbol is π ) times the diameter.

The radius is the distance from the center to the edge.

The diameter starts at one side of the circle, goes through the center and ends on

the other side. So the diameter is twice the radius.

Lines

A line that goes from one point to another on the

circle's circumference is called a chord.

If that line passes through the center it is called a

diameter.

If a line “just touches” the circle as it passes it is

called a tangent.

And a part of the circumference is called an arc.

Slices

There are two main “slices” of a circle

The “pizza” slice is called a sector.

And the slice made by a chord is called a segment.

http://www.mathsisfun.com/geometry/circle-sector-segment.html











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2. POINTS, LINES AND PLANES IN SPACE.

Euclid defined:

The point has no dimension. It is usually represented by a dot.

A line has one dimension: depth.

A plane has two dimensions: width and depth.

Solid Geometry is the geometry of three-dimensional space, the king of space

we live in three dimensions. It is called three-dimensional (3D) because there are three

dimensions: width, depth and height.

2.1. POSITION OF TWO LINES IN SPACE.

Two lines in space can be:

Two lines that meet at a point are called intersecting lines.

Two parallel lines do not intersect at any point.

Two lines that have all their points in common are called coincident lines.

Two lines which have no intersections but are not parallel are called skew

lines.

INTERSECTING LINES PARALLEL LINES

COINCIDENT LINES SKEW LINES




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2.2. POSITION OF A LINE AND A PLANE IN SPACE.

A line and a plane in space can be:

The line is contained within the plane.

The line is parallel to the plane.

The line cuts through the plane at a certain point.

2.3. THE POSITION OF TWO PLANES.

Two planes in space can be:

Two planes that meet at a line are called intersecting planes.

Two planes that have all their points in common are called coincident planes.

Two parallel planes do not intersect at any line.

INTERSECTING COINCIDENT PARALLELPLANES PLANES PLANES




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Particular case of intersecting planes:

Two planes are perpendicular if they form a right angle.

http://evamate.blogspot.com/2009/02/videos-para-geometria-2-bach-ccnn.html

2.4. DIHEDRAL ANGLE.

A dihedral angle is an angle formed by two semi-planes.

3. POLYHEDRAL.

The houses we live in, the room we are in now, books, pieces of furniture...

almost everything has a polyhedral shape: we live surrounded by polyhedral.

A polyhedron is a solid figure bounded by flat polygonal faces.







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3.1. ELEMENTS OF A POLYHEDRON.

The elements of a polyhedron are:

Faces: The faces of a polyhedron are each of the two dimensional polygons

that border the polyhedron.

Edges: The edges of a polyhedron are the sides of the faces of the polyhedron.

Two faces have an edge in common.

Vertices: The vertices of a polyhedron are the vertices of each of the faces of

the polyhedron. Three faces coincide with the same vertex.

Dihedral angles: The dihedral angles are formed between two faces of all

neighboring polygons.

Polyhedral angles: Polyhedral angles are formed by three or more faces of the

polyhedron and have a common vertex.

Diagonals: The diagonals of a polyhedron are the segments joining two

vertices not belonging to the same face.




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3.2. UNFOLD OF POLYHEDRAL.

A polyhedron is formed by extending the faces of the polyhedron on the same

plane.

4. REGULAR POLYHEDRA OR PLATONIC SOLIDS.

Regular polyhedral satisfy these conditions: The faces of regular polyhedra are equal regular polygons.

The same number of faces with the same shape converge in all the vertices,

that is, they form equal polyhedral angles. So, their polyhedral angles are equal.

The dihedral angles that form two border faces are equal. So, their dihedral

angles are equal.

http://www.kalipedia.com/kalipediamedia/matematicas/media/200709/26/geometria/20070926klpmatgeo_388.Ges.SCO.png




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There are only five regular polyhedra. They are called Platonic solids. They are

the tetrahedron, the octahedron, the icosahedron, the hexahedron or cube and the

dodecahedron.

Let’s see:

http://www.iessandoval.net/sandoval/aplica/activi_mate/actividades/poliedros/mar

co_poliedros.htm

A regular tetrahedron is a regular polyhedron composed of 4 equally sized

equilateral triangles.

A regular octahedron is a regular polyhedron composed of 8 equal equilateral

triangles.

A regular icosahedron is a regular polyhedron composed of 20 equally sized

equilateral triangles.

http://www.iessandoval.net/sandoval/aplica/activi_mate/actividades/poliedros/marco_poliedros.htm








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A cube or regular hexahedron is a regular polyhedron composed of 6 equal

squares.

A regular dodecahedron is a regular polyhedron composed of 12 regular

pentagons.

In Geometry an Archimedean solid is a highly symmetric, semi-regular convex

polyhedron composed of two or more types of regular polygons meeting in identical

vertices. They are distinct from the Platonic solids, which are composed of only one

type of polygon meeting in identical vertices.

There are 13 Archimedean Solids. (homework )

See the pages:

Paper models of polyhedral: http://www.korthalsaltes.com/index.html

Papercraft: http://www.4kids.tv/papercraft

http://www.korthalsaltes.com/index.html



http://www.4kids.tv/papercraft








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5. EULER’S FORMULA.

It is verified that in all convex polyhedra:

Number of vertices + Number of faces - Number of edges = 2

E. d.:

V + F – E = 2

Let's try with the 5 Platonic solids:

Name Faces Vertices Edges F+V-E

Tetrahedron 4 4 6 2

Cube 6 8 12 2

Octahedron 8 6 12 2

Dodecahedron 12 20 30 2

Icosahedron 20 12 30 2




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6. PRISMS.

A prism is a polyhedron with an n -sided polygonal base, a translated copy (not

in the same plane as the first), and n other faces (all parallelograms) joining

corresponding sides of the two bases.

BE CAREFUL: NO CURVED SIDES!

For example, a cylinder is not a prism because it has curved sides.

A right prism is a prism in which the joining edges and faces are perpendicular to

the base faces. This applies if the joining faces are rectangular.

If the joining edges and faces are not perpendicular to the base faces, it is called

an oblique prism.

RIGHT PRISM OBLIQUE PRISM




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6.1. TYPES OF PRISMS.

Prisms are named according to their base, so a prism with a pentagonal base is

called a pentagonal prism.

NAME BASETRIANGULAR PRISM

PENTAGONAL PRISM

6.2. SURFACE AREA OF A RIGHT PRISM.

The lateral surface area L (area of the vertical sides only) of any right prism is

equal to the perimeter of the base times the height of the prism.

The base surface area B is the area of the base.

The total surface area T of any right prism is equal to two times the area of the

base plus the lateral surface area.

T = 2B + Ph




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Example. Calculate the total surface area of this triangular prism.

h = 3 cm

= ∙ .

=.

= .

Perimeter of base = 1.5 + 1 + 2 = 4.5 cm

Lateral surface area = L = Perimeter ∙ height = 4.5 ∙ 3 = 13.5 cm2

Total surface area = T = 2B + L = 2 ∙ 0.75 + 13.5 = 1.5 + 13.5 = 15 cm2

7. PARALLELEPIPEDS.

A parallelepiped is a prism of which the base is a parallelogram.

7.1. TYPES OF PARALLELEPIPEDS.

A right parallelepiped is a parallelepiped in which the joining edges and faces

are perpendicular to the base faces. This applies if the joining faces are

rectangular.

If the joining edges and faces are not perpendicular to the base faces, it is

called an oblique parallelepiped.

RIGHT OBLIQUE

PARALLELEPIPED PARALLELEPIPED




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7.2. CUBOIDS.

A right parallelepiped is also called a cuboid or informally a rectangular box.

So, a cuboid is a parallelepiped that has all rectangular faces joined at right

angles.

Cuboids are very common in the world, from boxes to buildings we see them

everywhere.

A box with a

slot cut as a

handle

Cuboids in a cuboid

roomBoxes for model trains Now that's just silly!

The lateral surface area L of a cuboid is equal to the perimeter of the base times

the height of the cuboid. So, L = (2l + 2w) ∙ h

The base surface area B is B = lw

The total surface area T of a cuboid is equal to two times the area of the base

plus the lateral area. So, T = 2B + L

http://www.vitutor.com/geometry/solid/prisms.html#pa

http://www.vitutor.com/geometry/solid/prisms.html#pa




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Example. Calculate the surface area of a this parallelepiped.

7.3. CUBES.

A cube is a right parallelepiped which faces are squares.

Consider a cube with edge of a units. Then the total surface area of the cube is the

area of all the 6 faces that is 6a2.

Surface Area of a Cube = 6a2

Example. Calculate the surface area of this cube.

h = 8

B = 5 ∙ 4 = 20

Perimeter of base = 2 ∙ 5 + 2 ∙ 4 = 10 + 8 = 18

Lateral area = L = Perimeter ∙ height = 18 ∙ 8 =144 u2

Total area = T = 2B + L = 2 ∙ 20 + 144 = 40 + 144 = 184 u2

A = 6 ∙ 5 = 6 ∙ 25 = 150 cm




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7.4. DIAGONAL OF A CUBOID.

First, we calculate d. We use the Pytagoras’ Theorem.

Then, we calculate D. we use the Pytagoras’ Theorem.

Diagonal of a cuboid is given by:

= + +

Example. Calculate the diagonal of a cuboid with a length of 10 cm, width of 4

cm and height of 5 cm.

d = a + b

D = d + c = (a + b ) + c

So, D2 = a2 + b2 + c2

D = 102 + 42+52 = 100 + 16 + 25 = 141 = 11.87




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8. PYRAMIDS.

A pyramid is a polyhedron, whose base can be any polygon and whose lateral

faces are triangles with a common vertex (apex of the pyramid).

8.1. PARTS OF A PYRAMID.

The height of the pyramid is the measure of the perpendicular line segment from the

apex to the base.

The apothem of a regular pyramid is the height of one of its lateral faces.

8.2. TYPES OF PYRAMIDS.The base of a regular pyramid is a regular polygon and its faces are equally

sized.

The base of an irregular pyramid is an irregular polygon, and as a result, its

faces are not equally sized.




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A right pyramid has isosceles triangles as its lateral faces and its apex lies

directly above the midpoint of the base.

An oblique pyramid does not have all isosceles triangles as its lateral faces.

Types of pyramids by their base:

Triangular pyramid: The base is a triangle.

Square pyramid: The base is a square.

Pentagonal pyramid: The base is a pentagon.

Hexagonal pyramid: The base is a hexagon.

http://www.vitutor.com/geometry/plane/triangles.html

http://www.vitutor.com/geometry/plane/pentagon.html

http://www.vitutor.com/geometry/plane/hexagon.html

http://www.vitutor.com/geometry/plane/hexagon.html

http://www.vitutor.com/geometry/plane/pentagon.html

http://www.vitutor.com/geometry/plane/triangles.html




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When we think of pyramids we think of the Great Pyramids of Egypt.

They are actually square pyramids ,

because their base is a square.

8.3. UNFOLD OF A PYRAMID.




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8.4. CALCULATION OF THE APOTHEM OF A PYRAMID.

The apothem of a pyramid can be calculated if the height of the pyramid and

apothem of the base are known. Apply the Pythagoras’ Theorem and calculate the

hypotenuse for the shaded right triangle:

8.5. CALCULATION OF THE LATERAL EDGE OF A PYRAMID.

The lateral edge of the pyramid can be calculated if the height of the pyramid and

the radius of the base (or radius of the circumscribed circle) are known. Apply the

Pythagoras’ theorem and calculate the hypotenuse of the shaded right triangle.

http://www.vitutor.com/geometry/plane/right_triangle.html

http://www.vitutor.com/geometry/plane/pythagorean_theorem.html









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8.6. SURFACE AREA OF A PYRAMID.

Lateral Surface Area of a Pyramid.

PB = Perimeter of the base

AL = ∙

Surface Area of a Pyramid.

AT = AL + AB

Example 1: Calculate the surface area of a square pyramid whose base edge is 10

cm and its height is 12 cm.




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Example 2. Calculate the surface area of a hexagonal pyramid whose base edge is

16 cm and its lateral edge is 28 cm.




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9. TRUNCATED PYRAMID.

A truncated pyramid is the result of cutting a pyramid by a plane parallel to the

base and separating the part containing the apex.

9.1. ELEMENTS OF A TRUNCATED PYRAMID.

The lateral faces of a truncated pyramid are trapezoids.

The height of a truncated pyramid is the perpendicular distance between the

bases.

The apothem is the height of any of its lateral faces.

B1= Larger base

B2 = Smaller base

h = Height

http://es.wikipedia.org/wiki/Archivo:Frustum_with_symbols.svg




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9.2. UNFOLD OF A TRUNCATED PYRAMID

9.3. CALCULATION OF THE APOTHEM OF A TRUNCATED PYRAMID.

To calculate the apothem of a truncated pyramid, the height, the apothem of the

biggest base and the apothem of the minor base must be known. Apply the Pythagoras’

Theorem to determine the length of the hypotenuse of the red triangle to obtain the

apothem. I use the Pytagoras’ Theorem:









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9.4. SURFACE AREA OF A TRUNCATED PYRAMID.

Example: Calculate the lateral surface area and total surface area of the truncated

square pyramid whose larger base edge is 24 cm, smaller base edge is 14 cm and the

apothem of the truncated pyramid is 13 cm.




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10. EXERCISES AND PROBLEMS.

1. Write the name of the following polyhedral:

2. Complete this table:

Name Faces Vertices Edges

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

3. Write the elements of this polyhedron:




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4. Write the relative position of these elements in space

5. A box of chocolate has a triangular prism form. The base has form of an

equilateral triangle whose side is 10 cm. The height of the box is 25 cm.

Calculate:

a)

The lateral area of the box. b) The total surface area of the box.

6. Unfold the following polyhedron:

7. Calculate the diagonal of a cuboid with a length of 12 cm, width of 6 cm and

height of 7 cm.

8. We need to build a box with a lid whose base is rectangular and which

measures 2 cm wide by 30 cm long and has a height of 10 cm. How muchcardboard do we need?

9. Unfold a hexahedron (cube) whose edge is 3 cm long. Calculate its area.

10. What is the surface area of a cube of edge length 5 inches?




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11. Calculate the surface area of the following prism:

12. What is the surface area of a cube of edge length 4 cm?

13. What is the surface area of the cuboid (rectangular prism)?

14. What is the surface area of the cuboid (rectangular prism)?

15. Find the surface area of this cuboid.

16. Find the surface area of a right, regular pyramid with a hexagon at the base,

if the base edge has length 6 cm and the side edge has length 12 cm.




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17. Find the surface area of this square prism.

18. A cube has lengths of 9 inches. Find its surface area.

19. Find the surface area of a tetrahedron, if the edge has length 8 cm.

20. Calculate the lateral surface area and total surface area of the truncated

square pyramid with the following characteristics:

Bigger base = square whose side is 4 cm long.

Minor base = square whose side is 2 cm long.

Lateral faces = isosceles trapezium. The height of the isosceles trapezium

is 25 cm.




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11. SELF EVALUATION.

1. Calculate the total surface area of a hexagonal regular prism with a height of

6 cm. The side of the base is 4 cm long and the apothem is 3.5 cm long.

2. Find the surface area of this cuboid.

3. Look at the figure and answer:

a) The point A is …

b) AB is … c) AH is …

d) CD is …

4. A pyramid with a regular hexagonal base measuring 8 cm long and a height

of 12 cm.

a) Unfold the pyramid.

b) Calculate the height of the isosceles triangles of the faces.

5. The bases of a truncated pyramid are square-shared and its sides are 2 cm

and 6 cm long. The height is 10 cm.

a) Calculate the area of a face.

b) Calculate the total surface area.




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6. Look the figures and complete the table:

Figure 1 Figure 2 Figure 3

Name Faces Vertices Edges

Figure 1

Figure 2

Figure 3

7. Write the relative position of these elements in space:

8. Unfold a tetrahedron whose edge is 4 cm long. Calculate its area.

9. Unfold an octahedron whose edge is 3.5 cm long. Calculate its area.

10. Calculate the diagonal of a cuboid with a length of 15 cm, width of 9 cm and

height of 10 cm.




HOMEWORK 1. ARCHIMEDEAN SOLIDS

Find information about Archimedean solids. The homework has the following

index:

1. Introduction. Origin of name.

2. Archimedean Solids. Classification and definitions.

3. Euler’s Formula with Archimedean Solids.

4. Construction of Archimedean Solids.

5. Archimedean Solids in the world.

6. Conclusion.

7. Bibliography and webgraphy.

Then, each group will present the homework in class.

HOMEWORK 2. SALVADOR DALÍ

Find information about the painter Salvador Dali.

Then find information about the pictures “Crucifixion” and “Last supper ”.

Look at the pictures and describe them.

How are they constructed?

Do you use a polyhedron? What?

What is the significance of the polyhedra in these pictures? Make a small

analysis.

unit 8. polyhedra (6)

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