Unit Conversions (Part 1):

Units and Dimensions

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 when we convert between units, we have to keep the dimensions the same.

Video Source (08:45 mins) | Transcript

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 that an object has.

Perimeter

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

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

\(\text{Perimeter} = 1\:\text{m} + 1\:\text{m} + 1\:\text{m} + 1\:\text{m} = 4\:\text{m}\)

Area

Area is the measurement of the surface of a shape.

A square that measures 1 m by 1 m has an area of \(1\:\text{m}^{2}\) (pronounced meter squared or square meter).

\(\text{Area} = 1\:\text{m} \times 1\:\text{m} = 1\:\text{m}^{2}\)

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, \(\text{meters}\times\text{meters}=\text{meters}^{2}\).

Volume

Volume is the amount of space that an object occupies.

A cube that measures 1 m by 1 m by 1 m has a volume of \(1\:\text{m}^{3}\) (pronounced meter cubed or cubic meter).

\(\text{Volume} = 1\:\text{m} \times 1\:\text{m} \times 1\:\text{m} = 1\:\text{m}^{3}\)

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, \(\text{meters}\times\text{meters}\times\text{meters}=\text{meters}^{3}=\text{m}^{3}\).

Summary

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

2-D: Two directional measurements are in \(\text{units}^{2}\), or square units, or units squared.

3-D: Three directional measurements are in \(\text{units}^{3}\), or cubic units, or units cubed.

Example

Here is a rectangular 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.

\(25 \:\text{ft} + 25 \:\text{ft} + 15 \:\text{ft} + 15 \:\text{ft} = 2(25 \:\text{ft}) + 2(15 \:\text{ft}) = 50 \:\text{ft} + 30 \:\text{ft} = 80 \:\text{ft}\)

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

Area = the amount of surface the land takes up

\((25 \:\text{ft})(15 \:\text{ft}) = 375 \:\text{ft}^{2}\)

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.

\(25 \:\text{ft} \times 15 \:\text{ft} \times 7 \:\text{ft} = 2625 \:\text{ft}^{3}\)

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

Additional Resources

• Khan Academy: U.S. Customary and Metric Units (5:09 mins; Transcripts)