The purpose of taking measurements is to eventually use them for some type of application. A number of the trades fields require the use of measurements as well as the topic of this section which is area, volume, and capacity.
Area: Is how much flat space an object has. (2-Dimensional) Think about the amount of paint to cover a wall.
Volume: how much space an object holds. (3-Dimensional) Think about how much concrete it would take to fill an old swimming pool, or the amount of mulch needed for a flower bed.
Capacity: A form of volume pertaining to the amount of liquid a container can hold.
| Shape Name | Rectangle/Square | Circle | Cylinder |
|---|---|---|---|
| Shape | |||
| Formula | Area = $l*w$ | Area=$\pi r^2$ | Volume = $\pi r^2h$ |
| Shape Name | Box | Sphere | Cone |
|---|---|---|---|
| Shape | |||
| Formula | Volume =$l * w * h$ | Volume = $4/3 π r³$ | Volume = $1/3 π r² h$ |
If a 12 inch diameter pizza requires 10 ounces of dough. How much dough is needed for a 16 inch pizza?
To answer this question, we need to consider how the weight of the dough will scale to the size of the pizza. The amount of dough needed is an example of volume. However, since both pizzas will be about the same thickness, the weight will scale with the area of the top of the pizza. We can find the area of each pizza using the formula for a circle:
$$A = \pi r^2$$
A 12" pizza would have a radius of 6" (12 / 2), giving us an area of: $$A = \pi r^2 = \pi(6)^2 = 36\pi \approx 113.04 \text{ in}^2$$
A 16" pizza would have a radius of 8" (16 / 2), giving us an area of: $$A = \pi r^2 = \pi(8)^2 = 64\pi \approx 200.96 \text{ in}^2$$
Notice that if both pizzas were 1 inch thick, their volumes would be 113 in³ and 201 in³ respectively, which is the same ratio as the area. Since the thickness is the same for both pizzas, we can safely ignore the thickness, since it is unknown.
Using proportional reasoning that was learned in Chapter 5.1, we can make a proportion of the amount of dough to the area of the pizza:
\[\frac{10\text{ oz}}{113\text{ in}^2} = \frac{x\text{ oz}}{201\text{ in}^2}\]\[201 \cdot 10 = 113 \cdot x\] \[2010 = 113 x\]\[x = \frac{2010}{113}\]\[x \approx 17.788\]
This means that for a 16 inch pizza we'd need approximately 18 oz of dough.
Your grandmother has asked you to help her with her flower beds. She has one flower bed as outlined in the figure below. If she wants to apply mulch that is 3 inches deep. How many cubic yards of mulch does she need?
First we need to "cut" the shape into two shapes that we are able to find the area of. There are multiple ways this could be done. We are going to cut this into two trapezoids. But you could opt to cut it into rectangles. At this time we are also going to label each shape for ease of identity.
The area of a trapezoid is:
$$A = \frac{1}{2}(b_1+b_2) h$$
Looking at T1 we can see that we are missing one of the bases and the vertical height. Using simple math we can locate the height of this trapezoid to be:
$$15 - 7 = 8 \text{ ft}$$
and the second base:
$$15 - 8 = 7 \text{ ft}$$
It just happens that T1 is the same size trapezoid as T2. This means we can find the area of one trapezoid and then multiply it by 2.
This means the area of Grandma's flower bed is:
$$2 \cdot 88 = 176 \text{ sq ft}$$
Next we need the volume. Grandma wanted the mulch to be 3 inches thick. Since the area is in square feet, we convert the thickness to feet:
$$3 \text{ in} = \frac{3}{12} \text{ ft} = 0.25 \text{ ft}$$
Volume of the flower bed:
$$0.25 \cdot 176 = 44 \text{ cubic feet}$$
Mulch is sold by the cubic yard. Recall that:
$$1 \text{ yard} = 3 \text{ feet}$$
Cubing this to convert cubic measurements:
Now convert cubic feet to cubic yards:
$$\frac{44 \text{ ft}^3}{1} \cdot \frac{1 \text{ yd}^3}{27 \text{ ft}^3} = \frac{44}{27} \text{ yd}^3 \approx 1.63 \text{ yd}^3$$
If each bag of mulch covers 0.43 cubic yards, the number of bags needed is:
$$\frac{1.63}{0.43} \approx 3.79 \text{ bags}$$
This means you'll need to pick up 4 bags total to cover Grandma's flower bed.
There are endless ways to apply this information to problem-solving in the real world. It can be difficult to navigate at times, but breaking the problem into smaller pieces, like we did with trapezoids, helps simplify the calculations.
Kareem is looking to replace his old deck by upgrading it to a new deck that using composite decking. The decking he is interested in only comes in 12 foot lengths by 8 inch width. He needs to build the deck such that the boards run perpendicular with his house. His deck is 22 feet wide by 10 feet long.
How many boards does Kareem need to order?
To figure out how many boards Kareem needs, he needs to take the 22 feet and divide it by 8 in. However, we know that you have to have common units to do this correctly. At this point he could either convert the 22 feet into inches or convert the 8 in into feet. We will opt to convert 8 in into feet, which would be:
$$\frac{8}{12} = \frac{2}{3} \text{ foot}$$
Next, we take the 22 feet and divide it by the fraction:
$$\frac{22}{\frac{2}{3}} = \frac{22}{1} \cdot \frac{3}{2} = 33$$
Remember that dividing by a fraction is the same as multiplying by the reciprocal of the fraction (flip it over). According to the math, Kareem would need 33 boards to redo his deck.
Now let's talk a little about capacity. Capacity is the amount of liquid that an object can hold. The Metric system uses the measurement Liter (L) to measure liquids, while the English system uses the measurements below.
One thing to note about English units of capacity is that a fluid ounce is a measure of volume, it is not a weight of an object. For example, if you took 8 fluid ounces of milk and compared it to 8 fluid ounces of water, they would have different weights in ounces. We will discuss weight more in the next section.
| 1 ft³ ≈ 7.48 gal | 1 yd³ ≈ 202 gal | 1 gal ≈ 231 in³ |
| 1 Liter ≈ 1.06 qts | 1 mL ≈ 1 cm³ | 1 Liter ≈ 0.264 gal |
| 1 ft³ (freshwater) ≈ 62.5 lbs | 1 ft³ (saltwater) ≈ 64 lbs |
We know that there is 1 liter in 1.06 qts, and that there are 2 pts for every 1 qt. Using dimensional analysis:
$$\frac{1\ \text{pt}}{1} \cdot \frac{1\ \text{qt}}{2\ \text{pt}} \cdot \frac{1\ \text{L}}{1.06\ \text{qt}} = \frac{1}{2(1.06)} = \frac{1}{2.12} \approx 0.472\ \text{L}$$
Next, convert liters to milliliters using the metric ladder method (milli = 10⁻³):
$$0.472 \cdot 10^3 = 0.472 \cdot 1000 = 472\ \text{mL}$$
Meriam would need 472 mL of milk to make the recipe.
Charlie is looking to install a freshwater fish tank that holds 100 gallons of water. How many cubic feet of water does he need to fill the tank, and how much will the water weigh?
First, convert gallons to cubic feet:
$$\frac{100\ \text{gal}}{1} \cdot \frac{1\ \text{ft}^3}{7.48\ \text{gal}} \approx 13.37\ \text{ft}^3$$
Next, multiply by the weight of 1 cubic foot of freshwater (62.5 lbs):
$$13.37 \cdot 62.5 \approx 835.63\ \text{lbs}$$
Charlie will need approximately 13 cubic feet of water weighing approximately 836 lbs.
According to a study, an average of 2,476,000 cubic centimeters of Cola are consumed worldwide every second. How many kiloliters is that? How many gallons?
First, convert cubic centimeters to kiloliters (1 kL = 10⁶ mL, 1 cm³ = 1 mL):
$$\frac{2,476,000\ \text{cm}^3}{1} \cdot \frac{1\ \text{mL}}{1\ \text{cm}^3} \cdot \frac{1}{10^6} = \frac{2,476,000}{1,000,000} = 2.476\ \text{kL}$$
Next, convert liters to gallons (1 L ≈ 0.264 gal). First, convert cm³ to L by moving up 3 steps (divide by 10³):
$$\frac{2,476,000\ \text{cm}^3}{1} \cdot \frac{1\ \text{mL}}{1\ \text{cm}^3} \cdot \frac{1}{10^3} \cdot \frac{0.264\ \text{gal}}{1\ \text{L}} = 656.14\ \text{gal}$$
This means that on average, 656 gallons of Cola are consumed every second.