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Units and Dimensions
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First, we will talk about dimensional units, which will just give more detail into the units we started using last week. It’s important to remember how many dimensions we’re looking at because we have to keep the dimensions the same when we convert between units.

Units and Dimensions

### Dimensions

"Dimension" means to measure in one direction. A line only has one dimension because it is only measured in one direction.

A flat shape has two dimensions, and a 3-D object, like a cube, is called 3-D because it literally means that it can be measured in three directions.

There are many words that we use to express dimensional directions. Many can be used interchangeably.

Height, width, depth, length, and breadth are common words used to express measurements in different directions. There aren’t specific rules on what to call height versus what to call depth or width or length. Just know that these words represent different directions of an object.

### Perimeter

Perimeter means the measure of the outside boundary of a shape.

A square that measures 1m by 1m has a perimeter of 4m.

$$Perimeter = 1m + 1m + 1m + 1m = 4m$$

### Area

Area is the measurement of the surface of a shape.

A square that measures 1m by 1m has an area of $$1m^2$$ (pronounced meter squared or square meter).

$$Area = 1m \times 1m = 1m^2$$

In this case, the units of measurement for this square are in meters. Since area is calculated as length × width, the units are also in meters × meters. According to the rules of exponents, $$meters \times meters = meters^2 = m^2$$.

### Volume

Volume is the amount of space that an object occupies.

A cube that measures 1m by 1m by 1m has a volume of $$1m^3$$ (pronounced meter cubed or cubic meter).

$$Volume = 1m \times 1m \times 1m = 1m^3$$

In this case, the units of measurement for this cube are in meters. Since volume is calculated as length × width × height (or height × breadth × depth, or some other combination of words), the units for volume are also in meters × meters × meters. According to the rules of exponents, $$meters \times meters \times meters = meters^3=m^3$$.

### Summary

1-D: One directional measurement is in units without any exponents.

2-D: Two directional measurements are in $$units^2$$, or square units, or units squared.

3-D: Three directional measurements are in $$units^3$$, or cubic units, or units cubed.

### Example

Here is a little piece of land that measures 25 ft by 15 ft. In this case, our units are in feet (ft).

Perimeter: The distance around the piece of land.

$$25ft + 25ft + 15ft +15ft = 2(25ft) + 2(15ft) = 50ft + 30ft = 80ft$$

The units are in feet (ft) because we are only measuring lines.

Area: The amount of surface the land takes up.

$$(25ft) (15ft) = 375ft^2$$

The units for area are in square feet or feet squared. We could put 375 one foot by one foot squares on this piece of land.

Suppose we want to build a water tank on this piece of land that is 7 ft high.

Volume: The amount of space an object takes up.

$$25ft \times 15ft \times 7ft = 2625ft^3$$

The units for volume are cubed. We could put 2625 1 ft by 1 ft by 1 ft cubes in the space.

### Practice Problems

1. A rectangular postage stamp has a length of 21 mm and a width of 24 mm. Find the units for the perimeter of the postage stamp. (
Solution
Solution:
Perimeter deals with only one dimension (distance or length around the outside boundary), so our answer will be millimeters (mm).
)
2. A large piece of land is rectangular in shape and has a length of 32 miles and a width of 18 miles. Find the units for the perimeter of this piece of land. (
Solution
Solution:
Perimeter deals with only one dimension (distance or length around the outside boundary), so our answer will be miles (mi).
)
3. A rectangular room has a length of 10 m and a width of 8 m. Find the units for the area of the room. (
Solution
Solution:
Area deals with two dimensions, or two directional measurements (length × width), so our answer will be meters squared or $${\text{m}}^2$$.
)
4. A rectangular portrait measures 16 in by 12 in. Find the units for the area of the portrait. (
Solution
Solution:
Area deals with two dimensions, or two directional measurements (length × width), so our answer will be inches squared or $${\text{in}}^2$$.
)
5. A rectangular swimming pool has a length of 16 ft, a width of 12 ft, and a depth of 6 ft. Find the units for the volume of the swimming pool. (
Solution
Solution:
Volume deals with three dimensions, or three directional measurements (length × width × height), so our answer will be feet cubed or $${\text{ft}}^3$$.
)
6. A pizza box has a square top with two adjacent sides, both measuring 33 cm. The pizza box also has a depth of 5 cm. Find the units for the volume of the pizza box. (
Solution
Solution:
Volume deals with three dimensions, or three directional measurements (length × width × height), so our answer will be centimeters cubed or $${\text{cm}}^3$$.
)

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