Density mass volume triangle

Here we will learn about mass, density and volume, including what they are and how they are related to each other.

Students will first learn about the density mass volume triangle as part of algebra in high school.

What is the density mass volume triangle?

Mass, density and volume are physical properties of objects.

• \underline{Mass}

Mass is the measurement of the amount of matter there is in an object. The units of mass are usually kilograms (kg) or grams (g). The kilogram is the SI unit (the International System of Units) for mass.

• \underline{Volume}

Volume is a measure of how much space an object takes up. The volume of a substance is often measured in cubic centimeters ({cm}^3). The derived SI unit for volume is the cubic meter ({m}^3).

• \underline{Density}

Density is a compound measure made from the ratio of mass and volume. Density is a measure of the amount of matter there is per unit volume. The SI unit for density is kilograms per cubic meter (\mathrm{kg/m}^3 ).

Imagine there are two objects of the same size. The first object has high density, the second object has a lower density.

The object with high density would feel heavier than the object with the lower density. You could try this with two water bottles that are the same size. Fill one with water and fill the other with air.

The density of water is approximately 1,000\mathrm{~kg/m}^{3}.

The density of air is approximately 1.225\mathrm{~kg/m}^{3}.

The density of water is greater than the density of air, so the bottle filled with water is heavier.

Density

To calculate the mass, density or volume of an object, use the formula:

\text{Density}=\cfrac{\text{Mass}}{\text{Volume}}

This can be written as a formula triangle:

where M is the mass, D is the density, and V is the volume of an object.

Using this triangle, you can derive the formulas:

• M=D\times{V}
• D=M\div{V}
• V=M\div{D}

Note: The relative density, or specific gravity is the relationship between the density of a substance in relation to the density of another referenced substance.

For example, water at 4℃ has a density of 1\mathrm{~kg/L}. Liquid mercury has a density of 13.6\mathrm{~kg/L} and so its specific gravity is 13.6 as 13.6\div{1}=13.6. Since both units are in \mathrm{kg/L}, specific gravity has no units.

Step-by-step guide: Density formula

What is the density mass volume triangle?

Common Core State Standards

How does this relate to high school math?

• Algebra – Creating Equations (HS-A.CED.A.4)
  Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V=IR to highlight resistance R.

• Number and Quantity – Quantities (HS-N.Q.A.1)
  Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How to find the density, mass or volume

In order to find density, mass or volume:

1. Write the formula with the correct subject.
2. Substitute known values into the formula and do the calculation.
3. Write down the solution including the units.

Density mass volume triangle examples

Example 1: calculating the density (g/cm³)

A sphere is made from gold. It has volume 3\mathrm{~cm}^{3} and mass 60\mathrm{~g}. Calculate the density of the object.

1. Write the formula with the correct subject.

To calculate the density, divide the mass of the sphere by the volume of the sphere.

2Substitute known values into the formula and do the calculation.

3Write down the solution including the units.

The units for mass here are grams \mathrm{g}.

The units for volume here are cubic centimeters, \mathrm{cm}^{3}.

As mass is divided by volume to get density, the units are \mathrm{g}\div\mathrm{cm}^{3}=\mathrm{g/cm}^{3}.

The density of the object is 20\text{ g/cm}^{3}.

Example 2: calculating the density (kg/m³)

An object is made from steel. It has a volume of 25\mathrm{~m}^{3} and a mass of 425\mathrm{~kg}. Calculate the density of the object.

Write the formula with the correct subject.

To calculate the density, divide the mass of the object by the volume of the object.

D=\cfrac{M}{V}

Example 3: calculating the volume (cm³)

The mass of an object is 600\mathrm{~g}. The density of the substance which the object is made from is 3.2\mathrm{~g/cm}^{3}. Calculate the volume of the object.

Write the formula with the correct subject.

To calculate the volume, we need to divide the mass of the object by the density of the substance.

V=\cfrac{M}{D}

Example 4: calculating the volume (m³)

The mass of an object is 1.5\mathrm{~kg}. The density of the substance which the object is made from is 6\mathrm{~g/m}^{3}. Calculate the volume of the object.

Write the formula with the correct subject.

To calculate the volume, divide the mass of the object by the density of the substance.

V=\cfrac{M}{D}

Example 5: calculate mass (g)

Copper has a density of 9\mathrm{~g/cm}^{3}. Calculate the mass of 240\mathrm{~cm}^{3} of copper.

Write the formula with the correct subject.

To calculate the mass, multiply the density of the substance by the volume of the object.

M=D\times{V}

Example 6: calculating the mass (kg)

Lead has a density of 11.29\mathrm{~g/cm}^{3}. Calculate the mass of 0.5\mathrm{~m}^{3} of lead. Write your answer in kilograms.

Write the formula with the correct subject.

To calculate the mass, multiply the density of the lead by the volume of the lead.

M=D\times{V}

Teaching tips for density mass volume triangle

• Have a unit conversion chart posted in the classroom for students to refer to.

• Remind students that the unit of volume can be a capacity unit, like gallons, but it can also be a mathematical unit like \mathrm{cm}^3.

• Let students experience density hands on, instead of only using practice problems worksheets. This can be as simple as pouring different liquids into the same container, observing what happens and making conjectures about the density of substances in the container.

• Have students copy a few example problems for them to refer to as they answer practice questions on their own.

• Since the focus of this skill is understanding the formula and the units, let struggling students use a density calculator until they are comfortable completing the calculations on their own.

Easy mistakes to make

• Forgetting how to find the volume
  You may need to find the volume of an object first. Remember, the volume of a rectangular prism is the product of its three dimensions.
  
  For example,
  
  
  \text{Volume}=6\times{3}\times{2}=36\mathrm{cm}^{3}

• Confusing the conversions of measurement
  Metric units are often used for mass, density and volume. It’s important to know how to convert between metric units.
  
  For example, knowing there are 100\mathrm{~cm} in 1\mathrm{~m}.

• Not paying attention to what the question is asking
  Check that you give the answer in the right form. You may have solved grams, but the answer needs to be in kilograms. You may have an answer which needs to be rounded to a specific number of significant figures or decimal places.

Related compound measures lessons

• Speed distance time
• Average speed formula
• Speed formula
• Density formula
• Mass from density and volume
• Pressure force area
• Pressure formula
• Flow rate
• Population density formula

The next lessons are

• Math Equations
• Math Formulas

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