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# Calculating Derived Quantities

## Measurements can be used in combination to describe physical objects in greater detail.

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Calculating Derived Quantities

This scientist is using a calculator. Doing science often requires calculations. Converting units—say from inches to centimeters—is one type of calculation that might be required. Calculations may also be needed to find derived quantities.

### What Are Derived Quantities?

Derived quantities are quantities that are calculated from two or more measurements. Derived quantities cannot be measured directly. They can only be computed. Many derived quantities are calculated in physical science. Three examples are area, volume, and density.

### Calculating Area

The area of a surface is how much space it covers. It’s easy to calculate the area of a surface if it has a regular shape, such as the blue rectangle in the sketch below. You simply substitute measurements of the surface into the correct formula. To find the area of a rectangular surface, use this formula:

Area (rectangular surface) = length × width (l × w)

Q: What is the area of the blue rectangle?

A: Substitute the values for the rectangle’s length and width into the formula for area:

Area = 9 cm × 5 cm = 45 cm2

Q: Can you use this formula to find the area of a square surface?

A: Yes, you can. A square has four sides that are all the same length, so you would substitute the same value for both length and width in the formula for the area of a rectangle.

### Calculating Volume

The volume of a solid object is how much space it takes up. It’s easy to calculate the volume of a solid if it has a simple, regular shape, such as the rectangular solid pictured in the sketch below. To find the volume of a rectangular solid, use this formula:

Volume (rectangular solid) = length × width × height (l × w × h)

Q: What is the volume of the blue rectangular solid?

A: Substitute the values for the rectangular solid’s length, width, and height into the formula for volume:

Volume = 10 cm × 3 cm × 5 cm = 150 cm3

### Calculating Density

Density is a quantity that expresses how much matter is packed into a given space. The amount of matter is its mass, and the space it takes up is its volume. To calculate the density of an object, then, you would use this formula:

Density=massvolume\begin{align*}\text{Density} = \dfrac{\text{mass}}{\text{volume}} \end{align*}

Q: The volume of the blue rectangular solid above is 150 cm3. If it has a mass of 300 g, what is its density?

A: The density of the rectangular solid is:

Density=300 g150 cm3=2 g/cm3\begin{align*}\text{Density} = \dfrac{300 \ \text{g}}{150 \ \text{cm}^3}= 2 \ \text{g/cm}^3 \end{align*}

Q: Suppose you have two boxes that are the same size but one box is full of feathers and the other box is full of books. Which box has greater density?

A: Both boxes have the same volume because they are the same size. However, the books have greater mass than the feathers. Therefore, the box of books has greater density.

### Units of Derived Quantities

A given derived quantity, such as area, is always expressed in the same type of units. For example, area is always expressed in squared units, such as cm2 or m2. If you calculate area and your answer isn’t in squared units, then you have made an error.

Q: What units are used to express volume?

A: Volume is expressed in cubed units, such as cm3 or m3.

Q: A certain derived quantity is expressed in the units kg/m3. Which derived quantity is it?

A: The derived quantity is density, which is mass (kg) divided by volume (m3).

### Summary

• Derived quantities are quantities that are calculated from two or more measurements. They include area, volume, and density.
• The area of a rectangular surface is calculated as its length multiplied by its width.
• The volume of a rectangular solid is calculated as the product of its length, width, and height.
• The density of an object is calculated as its mass divided by its volume.
• A given derived quantity is always expressed in the same type of units. For example, area is always expressed in squared units, such as cm2.

### Practice

1. Identify six fundamental units in physical science.
2. What is speed? How is it calculated? What are its SI units? (Hint: The symbol Δ represents a difference, or change, in a unit. For example, Δt represents a change in time.)
3. Which derived quantity equals force divided by area?

### Review

1. What is a derived quantity? Give an example.
2. What are the dimensions of a square that has an area of 4 cm2?
3. Explain how you would calculate the volume of a cube.
4. Which derived quantity is used to calculate density?
5. Which derived quantity might be measured in mm3?

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