## 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.

A **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 A**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_{L}:**P**and width the height of the prism_{B},**h.****Area of the bases A**This is the sum of the areas of the bases._{B}:

The **surface** **area of a right prism **is:

**A _{T} = A_{L} + 2·A_{B} = **

**P**

_{B}· h + 2·A_{B}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).

A **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.

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 A**It is the sum of the areas of the lateral faces but, if we look at the net, the lateral face is_{L}:**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:

** **

**A _{L}** =

**n · (b · a) =**(

**P**

_{B}· a)/2 where **P _{B}** the perimeter of the base, and

**a**the apothem of the pyramid.

**Area of the bases A**As the base is a regular polygon:_{B}:

** **

**A _{B}** = (

**P**

_{B}· a’)/2where **a’ **the apothem of the base.

The **surface** **area of a regular pyramid **is:

**A _{T} = A_{L} + A_{B} = **(

**P**

_{B}· a)/2 + P_{B}· a’)/2Now, 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

- You can get nets of prisms pyramids to make your own models: http://www.senteacher.org/wk/3dshape.php
- Names and parts of 3-dimensional figures
- Nets of 3-dimensional figures (You have to guess the solid when you are given the net)
- Surface area of prisms and cylinders
- Surface area of pyramids and cones

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