Archive for the ‘Lesson 10: Solids. Surface Areas’ Category

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Lesson 10: Solids. Surface Areas. An Extra Question with a Prize (IV)

May 23, 2012


Are you able to prove it?

Think of the net of a cone.



If the base of the cone has radius r and the cone has generatrix g , I told you the lateral area (the area of the circle sector) was πrg

Could you prove it?

There is an extra 0.5 in this lesson if you give the right proof. Anyway your interest and effort will be prized in the global mark of this term.

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Lesson 10: Solids. Surface Areas. Revolution solids (III)

May 22, 2012

5. Solids of revolution

A solid of revolution is a solid figure obtained by rotating a flat figure around a line that lies on the same plane. The axis of revolution is the line about which the revolution takes place.

5.1. Cylinder

A cylinder is a solid of revolution generated by a rectangle that rotates around one of its sides.

Elements of a cylinder

Surface area of a cylinder

The net of a cylinder is formed by:

  • A rectangle whose length is the length of the circumference of the bases and its width is the height of the cylinder.
  • Two equal circles that are its bases.

The surface area of a cylinder can be calculated from its net. So it will be the sum of the lateral area and the areas of the two equal bases:

  • Lateral area AL: It is the area of the rectangle whose length is the length of the circumference of the bases with radius r, 2πr and its width is the height of the cylinder,h.

AL = 2πr · h

  • Area of the bases AB: This is the sum of the areas of the bases with radius . 

AB = πr2

The surface area of a cylinder is:

AT = AL + 2·AB = 2πr · h + 2· πr2 = 2πr (h + r)

5.2. Cone

The cone is a solid of revolution generated by a right triangle that rotates around one of its legs.

Elements of a cone

Calculation of the generatrix of a cone

 By the Pythagorean Theorem, the slant height or generatrix (or slant height, s) of the cone is equal to:

Surface area of a cone

 The net of a cone is formed by:

  • Circular sector of length2πr · h, being r the radius of the base, and radius g the generatrix of the cone.
  • A circle that is the base

As in other cases we work out the surface area from its net. So it will be the sum of the lateral area and the area of the base:

  • Lateral area AL: It is the area of the circular sector whose length is 2πr and its radius is,g.

AL = πr · g

  • Area of the base AB: This is the area of the base with radius . 

AB = πr2

The area of a cone is:

AT = AL + AB = πr · g + · πr2 = πr (g + r)

You can get the net of  a cone and a cylinder: http://www.senteacher.org/wk/3dshape.php

5.3. Sphere

A spherical surface is the surface generated by rotating a circle about its diameter.

A sphere is the region of the space that is inside a spherical surface.

A sphere can also be seen as the region of the space obtained by rotating a semicircle around its diameter.

 Elements of a sphere

The centre is the interior point that is equidistant to all points on the surface of the sphere.

Radius is the distance from the centre to any point on the surface of the sphere, .

The diameter is the distance from one point through the centre to another point . A diameter is twice the radius.

Surface area of a sphere

The surface area of sphere of radius is :

A = 4 πr2

In the following links you will find exercises about all the 3-D shapes we studied:

Finally, the following video will show you how to find the areas of a cylinder, a cone and a sphere. Improve your listening!


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Lesson 10: Solids. Surface Areas. Prisms and Pyramids (II)

May 22, 2012

3. Prisms

Prisms are polyhedrons that have two parallel, equally sized faces called bases and their lateral faces are parallelograms.

prism is a 3D shape which has a constant cross-section – both ends of the solid are the same shape and anywhere you cut parallel to these ends gives you the same shape too.

3.1.Types of prisms

To name a prism we use to refer to the base polygon. This way we can name them by their bases:

Right prisms are those that have lateral faces rectangles or squares, i.e. basic edges are perpendicular to the lateral edges

Oblique prisms are those whose lateral faces are rhomboids.

Oblique prism

Regular prisms are right prism whose bases are regular polygons.

Irregular prisms are prisms whose bases are irregular polygons.

Parallelepipeds are prisms whose bases are parallelograms.

Cuboids are right parallelepipeds, i.e. their faces are rectangular. Cuboids are very common in daily life. They are rectangular prisms.

3.2. Elements of a prism

3.3. Surface area of a prism

The net of a right prism is formed by:

  • A rectangle of length all the sides of the base, this is the perimeter of the base, and width the height of the prism.
  • Two equal polygons, the ones that are the bases.

It is easy to deduce the surface area of a prism from its net. So it will be the sum of the lateral area and the areas of the two equal basis:

  • Lateral area AL: It is the sum of the areas of the lateral faces but, if we look at the net, the lateral surface is a rectangle of length the perimeter of the base PB, and width the height of the prism h.
  • Area of the bases AB: This is the sum of the areas of the bases.

The surface area of a right prism is:

AT = AL + 2·AB =  PB · h + 2·AB

Now, you can see a video about how to find the surface area of a prism.

4. 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).

pyramid has sloping sides that meet at a point.

4.1. Elements of a pyramid

4.2. Types of pyramids

To name a pyramid we use to refer to its base polygon. This way we can name them by their bases:



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 sides.

Oblique pyramid

A regular pyramid is right pyramid whose base is a regular polygon and its lateral faces are equally sized. In other case is named irregular pyramid.

4.3 Surface area of a regular pyramid

The net of a regular pyramid is formed by:

  • As many isosceles triangles as sides the base has.
  • The base polygon

Let  the number of sides of the base of the regular polygon that is the base of the pyramid.

We deduce the surface area of a prism from its net. So it will be the sum of the lateral area and the area of the base:

  • Lateral area AL: It is the sum of the areas of the lateral faces but, if we look at the net, the lateral face is n isosceles triangles whose base is the side of the polygon, b, of the base and height the apothem of the pyramid. If we add them up, we have:

 

AL = n · (b · a) = (PB · a)/2

 where  PB the perimeter of the base, and a the apothem of the pyramid.

  • Area of the bases AB: As the base is a regular polygon:

 

AB = (PB · a’)/2

where a’ the apothem of the base.

The surface area of a regular pyramid is:

AT = AL + AB =  (PB · a)/2 + PB · a’)/2

Now, you can see a video about how to find the surface area of a pyramid, in this case cuadrangular).

The following links will provide you enough exercises

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Lesson 10: Solids.Surface Areas (I)

May 16, 2012

Vocabulary on Solids 

Lesson 10: Solids. Surface Areas (Notes)

 Practice Problems on Classifying Solids

Practice problems on Surface Area of Solids

I Have to Know by the End of this Lesson

 

Three-D  shapes have 3-dimensions- length, width and depth. We are going to study some of them: polyhedrons and revolution solids.

1. Polyhedrons

A polyhedron is a three-dimensional region of the space bounded by polygons.

Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren’t polyhedrons).

If you click on the links you will learn more about these 3-D shapes in the web where it is taken from the table below, www.mathisfun.com. You can also find questions with answers.

  Polyhedrons :
(they must have flat faces)
tetrahedronhexahedronoctahedrondodecahedornicosahedron Platonic Solids
triangular prismsquare prismpentagonal prism Prisms
triangular pyramidsquare pyramidpentagonal pyramid Pyramids
     
  Non-Polyhedra:
(if any surface is not flat)
sphere Sphere torus Torus
cylinder Cylinder cone Cone

Elements of a polyhedron

Faces: These polygons that limit the polyhedrons.

Edges: Line segments where two faces of a polyhedron meet. They are sides of the faces.

Vertices of the polyhedron: Points at which three or more polyhedron edges of a polyhedron meet.

Diagonals of a polyhedron: Segments, joining two vertices, which are not placed on the same face, are called diagonals of polyhedron. The tetrahedron has no diagonals.

Dihedral angle (also called the face angle) is the internal angle at which two adjacent faces meet. All dihedral angles between the edges are ≤ 180º.

Polyhedral angle: is the portion of space limited by tree or more faces which meet at a vertex. All polyhedral angles between the edges are ≤ 360º .

Net: It is an arrangement of edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. There are several possibilities for a net of a polyhedron.

    Tetrahedrom                                                   Net  for a tetrahedron

You can find the net of the main polyhedron and their different possibilities on

http://gwydir.demon.co.uk/jo/solid/index.htm

2. Regular polyhedrons

2.1. Types of polyhedrons

A convex polyhedron is defined as follows: no line segment joining two of its points contains a point belonging to its exterior. There are many examples known by you: the cube, prisms, pyramids….

A concave polyhedron, on the other hand, will have line segments that join two of its points with all but the two points lying in its exterior.

Below is an example of a concave polyhedron.

The study of polyhedrons was a popular study item in Greek geometry even before the time of Plato (427 – 347 B.C.E.) In 1640, Rene Descartes, a French philosopher, mathematician, and scientist, observed the following formula. In 1752, Leonhard Euler, a Swiss mathematician, rediscovered and used it.

In a convex polyhedron:

If  F = number of faces , V = number of vertices and E = number of edges, then

 F + V = E +2

This formula is named Euler’s Formula.

All the convex polyhedrons verify this formula. There are some concave polyhedrons that verify it. However, there are concave polyhedrons that don’t verify this formula.

2.2. Regular polyhedrons

A regular polyhedron is a polyhedron where:

  • each face is the same regular polygon
  • the same number of faces (polygons) meet at each vertex (corner)

They are also called Platonic solids.

There are five Platonic solids:

1. Tetrahedron, which has three equilateral triangles at each corner.
2. Cube, which has three squares at each corner.
3. Octahedron, which has four equilateral triangles at each corner.
4. dodecahedron, which has three regular pentagons at each corner.
5. Icosahedron, which has five equilateral triangles at each corner.

Exercise: Fill in this table and check they satisfy Euler’s formula. Why?

                faces edges vertices
tetrahedron      ___   ___    ___
cube             ___   ___    ___
octahedron       ___   ___    ___
dodecahedron     ___   ___    ___
icosahedron      ___   ___    ___