6. Geometry & Measurement: Solids

Lesson

Area is the number of square units needed to cover a surface or figure and relates to a 2D object. The surface area is the area covering a 3D object.

Surface area has lots of applications. Here are some examples:

- In manufacturing we may need to calculate the cost of making boxes or sheet metal parts.
- In construction, surface area affects planning (how much to buy) and costs (how much to charge) in connection with such things as sheetrock, shingles, and paint.

Many objects have complex shapes to increase their surface area: the inside of your lungs, intestines, and brain; air purifiers, or radiators.

We will start by looking at how to find the surface area of a rectangular prism.

Rectangular prisms have three pairs of congruent faces. We can see below how we could break the rectangular prism above into three pairs of congruent rectangles. To find the total surface area, we must add up the area of all of the faces.

$=$= | $+$+ | $+$+ |

Have a look at this interactive to see how to unfold rectangular prisms.

- Slide the "Open/Close" slider to see the prism fold and unfold
- Slide one of the "Dimensions" sliders and then Open/Close again

Consider the questions below.

- What type of shape is formed if all of the Dimension sliders are set to the same level?
- How many faces does every rectangular prism have?
- If you change only one of the Dimension sliders how many congruent faces are there?
- What is the benefit to unfolding and opening the prism when calculating the surface area?
- Instead of finding the area of each of the $6$6 faces, how could you find the surface area more quickly?

When needing to calculate the surface area (SA) of a prism we need to add up the areas of individual faces. Make sure not to miss any faces but also try to look for clever methods, like using the fact that $2$2 faces might have the same area.

While we are just looking at rectangular prisms for now, the concept below will help us in future lessons too.

General: Surface area of a prism

$\text{Surface area of a prism }=\text{Sum of areas of faces}$Surface area of a prism =Sum of areas of faces

If we are just looking at a rectangular prism, we can use a formula instead of adding up all $6$6 faces separately.

As we saw with the applet above, there are three pairs of congruent rectangles.

- The top and bottom which are both $l\times w$
`l`×`w` - The left and right which are $l\times h$
`l`×`h` - The front and back which are $w\times h$
`w`×`h`

Since there are two of each of these rectangles we get the formula below.

Rectangular prism surface area formula

$\text{S.A. }=2lw+2lh+2wh$S.A. =2`l``w`+2`l``h`+2`w``h`

Consider the following cube with a side length equal to $6$6 cm. Find the total surface area.

Consider the following rectangular prism with length width and height equal to $12$12 m, $6$6 m and $4$4 m respectively.

Find the surface area of the prism.

What is the surface area of a cube with side length $4$4 cm?

The volume of a three dimensional shape is the amount of space that is contained within that shape.

A quantity of volume is represented in terms of the volume of a unit cube, which is a cube with side length $1$1 unit. By definition, a single unit cube has a volume of $1$1 cubic unit, written as $1$1 unit^{3}.

The image below shows a rectangular prism with length $5$5 units, width $3$3 units, and height $2$2 units. Notice that the length of each edge corresponds to the number of unit cubes that could be lined up side by side along that edge.

We can find the number of unit cubes that could fit inside the rectangular prism by taking the product of the three side lengths. This gives $5\times3\times2=30$5×3×2=30, so there are $30$30 unit cubes in the prism, which means it has a volume of $30$30 unit^{3}.

Use the sliders to change the length, width, and height of the rectangular prism. Consider the questions below.

- Why do you think all of the unit cubes in the base are shown?
- If we count the number of unit cubes in the base, how can we use the height to get the total volume (number of unit cubes)?
- What product could we use to find the volume?

In the same way that the area of a two dimensional shape is related to the product of two perpendicular lengths, the volume of a three dimensional shape is related to the product of three mutually perpendicular lengths (each of the three lengths is perpendicular to the other two).

Volume of a rectangular prism

The volume of a rectangular prism is given by

$\text{Volume }=\text{length }\times\text{width }\times\text{height }$Volume =length ×width ×height , or

$V=l\times w\times h$`V`=`l`×`w`×`h`

Use the three sliders for length, width, and height to see how changing these affect the rectangular prism. Click the boxes to see the formula and volume revealed.

A cube can be thought of as a special type of rectangular prism, one that has all sides equal in length. The formula for the volume of a cube is similar to the formula for the area of a square.

Volume of a cube

The volume of a cube is given by

$\text{Volume }=\text{side }\times\text{side }\times\text{side }$Volume =side ×side ×side , or

$V=s\times s\times s=s^3$`V`=`s`×`s`×`s`=`s`3

Find the volume of the following rectangular prism.

**Think**: The side lengths have units of cm, so the volume will be in cm^{3}.

**Do**: The base of the prism has a width of $2$2 cm and a length of $7$7 cm, and the height of the prism is $9$9 cm. We will use these sides in the formula for the volume of a rectangular prism.

$\text{Volume }$Volume | $=$= | $\text{length }\times\text{width }\times\text{height }$length ×width ×height | (Formula for the volume of a rectangular prism) |

$=$= | $7\times2\times9$7×2×9 | (Substitute the values for the length, width, and height) | |

$=$= | $126$126 | (Perform the multiplication to find the volume) |

So this rectangular prism has a volume of $126$126 cm^{3}.

The local swimming pool is $25$25 m long. It has eight lanes, each $2$2 m wide, and its depth is $1.5$1.5 m. What is the volume of water in the pool?

**Think**: The water in the pool is in the shape of a rectangular prism, so to find its volume we need to find the side lengths of this prism. The length and depth of the pool are two side lengths we can use. The final side length is found by multiplying the number of lanes by the width of each lane.

**Do**: First we calculate the width of the pool using the width of each swim lane: $8\times2$8×2 m $=16$=16 m. Next we use the formula for the volume of a rectangular prism.

$\text{Volume }$Volume | $=$= | $\text{length }\times\text{width }\times\text{height }$length ×width ×height | (Formula for the volume of a rectangular prism) |

$=$= | $25\times16\times1.5$25×16×1.5 | (Substitute the values for the length, width, and height) | |

$=$= | $600$600 | (Perform the multiplication to find the volume) |

So the water in the pool has a volume of $600$600 m^{3}.

**Reflect**: Even though the volume formula uses the terms "length", "width", and "height", when referring to everyday objects it may be more appropriate or more common to use alternative words like "width", "depth", or "thickness". In this example, we could just as well have used the formula $\text{Volume }=\text{length }\times\text{width }\times\text{depth }$Volume =length ×width ×depth .

We use special units to describe volume, based on the notion of cubic units described above. Because the units for length include inches, centimeters, meters, yards and miles we end up with the following units for volume.

Units of Volume

**cubic inches = in ^{3}**

(picture a cube with side lengths of $1$1 inch each - about the size of linking cubes)

**cubic centimeters = cm ^{3}**

(picture a cube with side lengths of $1$1 cm each - about the size of a dice)

**c****ubic yards = yd ^{3}**

(picture a cube with side lengths of $1$1 yd each - what could be this big?)

Before we start a question, it is important to check that all of the sides are in the **same unit**. If they aren't, then we should convert them to the same unit.

Find the volume of the rectangular prism shown.

Find the volume of the cube shown.

A box is $1$1 m long, $20$20 cm high and $30$30 cm wide. What is the volume of the box in cubic centimeters?

Describe how changing one measured attribute of a rectangular prism affects the volume and surface area