6.3 Geometry  (Page 2/7)

Volume of Prisms

Calculate the area of the base and multiply by the height to get the volume of a prism.

Volume

1. Write down the formula for each of the following volumes:

   

2. Calculate the following volumes:

   

3. A cube is a special prism that has all edges equal. This means that each face is a square. An example of a cube is a die. Show that for a cube with side length $a$ , the surface area is $6a^2$ and the volume is $a^3$ .

   

Now, what happens to the surface area if one dimension is multiplied by a constant? For example, how does the surface area change when the height of a rectangular prism is divided by 2?

The size of a prism is specified by the length of its sides. The prism in the diagram has sides of lengths $L$ , $b$ and $h$ .

1. Consider enlarging all sides of the prism by a constant factor $x$ , where $x>1$ . Calculate the volume and surface area of the enlarged prism as a function of the factor $x$ and the volume of the original volume.
2. In the same way as above now consider the case, where $0<x<1$ . Now calculate the reduction factor in the volume and the surface area.

1. Identify

   The volume of a prism is given by: $V=L\times b\times h$

   The surface area of the prism is given by: $A=2\times (L\times b+L\times h+b\times h)$

2. Rescale

   If all the sides of the prism get rescaled, the new sides will be:

   $$\begin{array}{ccc}\hfill L^{\text{'}}& =& x\times L\hfill \\ \hfill b^{\text{'}}& =& x\times b\hfill \\ \hfill h^{\text{'}}& =& x\times h\hfill \end{array}$$

   The new volume will then be given by:

   $$\begin{array}{ccc}\hfill V^{\text{'}}& =& L^{\text{'}}\times b^{\text{'}}\times h^{\text{'}}\hfill \\ & =& x\times L\times x\times b\times x\times h\hfill \\ & =& x^3\times L\times b\times h\hfill \\ & =& x^3\times V\hfill \end{array}$$

   The new surface area of the prism will be given by:

   $$\begin{array}{ccc}\hfill A^{\text{'}}& =& 2\times (L^{\text{'}}\times b^{\text{'}}+L^{\text{'}}\times h^{\text{'}}+b^{\text{'}}\times h^{\text{'}})\hfill \\ & =& 2\times (x\times L\times x\times b+x\times L\times x\times h+x\times b\times x\times h)\hfill \\ & =& x^2\times 2\times (L\times b+L\times h+b\times h)\hfill \\ & =& x^2\times A\hfill \end{array}$$
3. Interpreting the above results
   1. We found above that the new volume is given by: $V^{\text{'}}=x^3\times V$ Since $x>1$ , the volume of the prism will be increased by a factor of $x^3$ . The surface area of the rescaled prism was given by: $A^{\text{'}}=x^2\times A$ Again, since $x>1$ , the surface area will be increased by a factor of $x^2$ . Surface areas which are two dimensional increase with the square of the factor while volumes, which are three dimensional, increase with the cube of the factor.
   2. The answer here is based on the same ideas as above. In analogy, since here $0<x<1$ , the volume will be reduced by a factor of $x^3$ and the surface area will be decreased by a factor of $x^2$

When the length of one of the sides is multiplied by a constant the effect is to multiply the original volume by that constant, as for the example in [link] .

Polygons

Polygons are all around us. A stop sign is in the shape of an octagon, an eight-sided polygon. The honeycomb of a beehive consists of hexagonal cells. The top of a desk is a rectangle.

In this section, you will learn about similar polygons.

Similarity of polygons

Discussion : similar triangles

Fill in the table using the diagram and then answer the questions that follow.

1. What can you say about the numbers you calculated for: $\frac{\mathrm{AB}}{\mathrm{DE}}$ , $\frac{\mathrm{BC}}{\mathrm{EF}}$ , $\frac{\mathrm{AC}}{\mathrm{DF}}$ ?
2. What can you say about $\widehat{A}$ and $\widehat{D}$ ?
3. What can you say about $\widehat{B}$ and $\widehat{E}$ ?
4. What can you say about $\widehat{C}$ and $\widehat{F}$ ?

If two polygons are similar , one is an enlargement of the other. This means that the two polygons will have the same angles and their sides will be in the same proportion.