## 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