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Math 6 · G.MG.A.2

Density

A big log floats and a small pebble sinks, yet the log has far more mass. What does the pebble have more of?

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Volume of a boxGrade 5Metric mass unitsGrade 3Rate and unit rateGrade 6
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Flow rate
Density

Why this exists

Ask which is heavier, iron or wood, and the honest answer is “how much of each?” A pine log has more mass than an iron nail. Yet everyone feels that iron is the heavier material. What they mean is that iron packs more mass into the same amount of space.

Density turns that feeling into a number. It compares materials fairly by asking each one the same question: how much mass is in one unit of volume? It is a Rates and unit rates, like a speed or the Flow rate of a tap, and the same per-one reasoning that runs every unit rate runs this one.

Definition

The density of a material is its mass per unit of volume:

\text{density} = \frac{\text{mass}}{\text{volume}}

In letters, D = \dfrac{m}{V}. With mass in grams and volume in cubic centimeters, density is in grams per cubic centimeter, written \text{g/cm}^3. With kilograms and cubic meters it is in kilograms per cubic meter, \text{kg/m}^3. Grams and kilograms are Metric mass units. This lesson talks about mass, not weight; Mass is not the same as weight explains the difference.

A density of 2.5\ \text{g/cm}^3 says: every cubic centimeter of this material has a mass of 2.5 grams.

The same per-unit idea works on a map. Population density is the number of people per unit of area, usually people per square kilometer:

\text{population density} = \frac{\text{people}}{\text{area}}

It measures how crowded a place is, the way density measures how tightly mass is packed.

Intuition

Figure 1 shows a glass block 4 cm long, 3 cm deep and 2 cm tall. By the volume of a box, V = 4 \times 3 \times 2 = 24\ \text{cm}^3, so the block is made of 24 one-centimeter cubes.

4\text{ cm}
2\text{ cm}
3\text{ cm}
Figure 1Every small cube is 1\text{ cm}^3 of the same glass, so every small cube has the same mass.

Suppose the whole block has mass 60 g. The glass is the same all through, so the 24 small cubes share the mass equally: each has 60 \div 24 = 2.5 g. That one cube’s mass is the density, 2.5\ \text{g/cm}^3.

A second block of the same glass, twice as long, has twice the cubes and twice the mass, but each cube still has 2.5 g. Size changes the mass; it does not change the density. Density belongs to the material, not to the object.

That is how a big thing can be light for its size. A pine log has far more mass than an iron bolt, but each cubic centimeter of pine holds about 0.5 g and each cubic centimeter of iron about 7.9 g. The log has more mass because it has more cubic centimeters; the bolt is denser because each of its cubic centimeters holds more.

\text{cork }0.24
\text{pine }0.5
\text{glass }2.5
\text{granite }2.7
\text{iron }7.9
\text{water } 1
0
1
2
3
4
5
6
7
8
\text{g/cm}^3
Figure 2Rounded densities in grams per cubic centimeter. Bars that stop short of the dashed line belong to solids that float in water.

Figure 2 compares a few materials. Every density in this lesson is rounded: real wood, rock and glass vary from piece to piece.

Results

One fact, three forms. For a material of density D, a piece of volume V has mass m, and

D = \frac{m}{V} \qquad m = D \times V \qquad V = \frac{m}{D}

Floating. Fresh water has density about 1\ \text{g/cm}^3. A solid piece of one material floats in water if its density is less than 1\ \text{g/cm}^3, and sinks if it is more.

Changing units. 1\ \text{g/cm}^3 = 1{,}000\ \text{kg/m}^3.

Population density. People = population density \times area, and area = people \div population density.

Why it is true

The three forms. If the material is the same all through, each cubic centimeter has mass D grams. A piece of V cubic centimeters is V such cubes, so its mass is V groups of D: m = D \times V. Division undoes multiplication, so dividing m by V gives back D, and dividing m by D gives V, the number of D-gram cubes that make up m grams.

Changing units. A cubic meter is 100 cm by 100 cm by 100 cm, so it holds 100 \times 100 \times 100 = 1{,}000{,}000 cubic centimeters. At 1 g each that is 1{,}000{,}000 g, which is 1{,}000 kg.

Floating. This is a fact from physics, known as Archimedes’ principle: water pushes up on an object in it just as hard as it would push up to hold up the water the object pushes aside. A solid less dense than water has less mass than the same volume of water, so the push is enough to hold it up before it is fully under. A denser solid has more mass than the water it pushes aside, and it sinks.

Worked examples

Example 1

Density of a rock sample

At the geology club, Stanislav puts a rock sample on a scale: 135 g. He lowers it into a measuring jug of water and the level rises by 50 mL. A milliliter is the same volume as a cubic centimeter. Find the rock’s density.

Solution

The rock pushes aside its own volume of water, so its volume is 50\ \text{cm}^3. Density is mass divided by volume:

D = \frac{135}{50} = 2.7\ \text{g/cm}^3

Each cubic centimeter of the rock has a mass of 2.7 g. Granite is about 2.7\ \text{g/cm}^3, so granite is a good guess for what it is.

Example 2

A block with a fraction edge

In her woodworking shop, Waltraud cuts a pine block 12 cm by 8 cm by 2\tfrac{1}{2} cm. Its mass is 120 g. Find its density, and say whether it floats.

Solution

First the volume, V = lwh, with a fraction edge as in volume with fraction edges:

V = 12 \times 8 \times 2\tfrac{1}{2} = 96 \times 2\tfrac{1}{2} = 240\ \text{cm}^3

Then the density:

D = \frac{120}{240} = 0.5\ \text{g/cm}^3

Since 0.5 < 1, each cubic centimeter of the block has less mass than a cubic centimeter of water, so the block floats.

Example 3

Mass from density

Venla makes candles from beeswax, which has a density of about 0.96\ \text{g/cm}^3. Her mold holds 250\ \text{cm}^3. What mass of wax fills it?

Solution

Each cubic centimeter of wax has mass 0.96 g, and the mold holds 250 of them:

m = D \times V = 0.96 \times 250 = 240\ \text{g}

Check with the unit rate another way: 100\ \text{cm}^3 of wax is 96 g, and 250\ \text{cm}^3 is two and a half times as much, so 2.5 \times 96 = 240 g. The wax is less dense than water, so a finished candle floats.

Example 4

Volume from density

Urszula has a quartz crystal of mass 53 g. Quartz has a density of about 2.65\ \text{g/cm}^3. Find the crystal’s volume.

Solution

Each 2.65 g of quartz takes up one cubic centimeter, so the volume is the number of 2.65-gram pieces in 53 g:

V = \frac{m}{D} = \frac{53}{2.65} = \frac{5300}{265} = 20\ \text{cm}^3

Multiplying top and bottom by 100 clears the decimal without changing the quotient. Check: 2.65 \times 20 = 53.

Example 5

Kilograms per cubic meter

Oak has a density of about 750\ \text{kg/m}^3. Waltraud’s oak beam is 4 m long, \tfrac{1}{4} m wide and \tfrac{1}{5} m thick. Find its mass. Then write oak’s density in \text{g/cm}^3 and say whether oak floats.

Solution

The volume is in cubic meters, which matches the density:

V = 4 \times \tfrac{1}{4} \times \tfrac{1}{5} = \tfrac{1}{5}\ \text{m}^3
m = D \times V = 750 \times \tfrac{1}{5} = 150\ \text{kg}

Since 1\ \text{g/cm}^3 is 1{,}000\ \text{kg/m}^3, divide by 1{,}000: 750 \div 1{,}000 = 0.75\ \text{g/cm}^3. That is less than 1, so oak floats.

Example 6

Big and light, small and heavy

Tuomas has two club samples. A chunk of pumice has mass 180 g and volume 300\ \text{cm}^3. A piece of basalt has mass 90 g and volume 30\ \text{cm}^3. Which has more mass? Which is denser? Which floats?

Solution

The pumice has more mass: 180 g against 90 g, twice as much. Now the densities:

\begin{aligned} \text{pumice: } \frac{180}{300} &= 0.6\ \text{g/cm}^3 \\ \text{basalt: } \frac{90}{30} &= 3\ \text{g/cm}^3 \end{aligned}

The basalt is 3 \div 0.6 = 5 times as dense. Pumice is full of tiny air bubbles, so each of its cubic centimeters holds little rock. Its density is below 1, so it floats and the basalt sinks. More mass and more density answer different questions: mass is about the whole object, density about one cubic centimeter of it.

Example 7

Population density

Tuomas compares two districts on a map. Northside has 48{,}000 people on 12\ \text{km}^2. Riverside has 30{,}000 people on 5\ \text{km}^2. Which is more crowded?

Solution

Divide people by area for each, in people per square kilometer:

\begin{aligned} \text{Northside: } \frac{48{,}000}{12} &= 4{,}000 \\ \text{Riverside: } \frac{30{,}000}{5} &= 6{,}000 \end{aligned}

Riverside has fewer people but is more crowded: each square kilometer holds 6{,}000 people against 4{,}000. It is Example 6 again, with area in place of volume.

Boundaries

One density needs one material, the same all through. A rock with veins of another mineral has different densities in different places, and m \div V gives only the average over the whole piece.

Density shifts with heat and squeezing. Warm materials spread out a little, so the same mass takes more room, and a gas squeezed into a tank becomes much denser. The densities here are for room temperature and ordinary air.

The floating rule is for a solid piece of one material, in fresh water. A hollow object is judged by its whole mass divided by the whole volume it takes up, air included. Sea water is a little denser than fresh water, about 1.03\ \text{g/cm}^3, so the dividing line moves slightly. A solid of density exactly 1\ \text{g/cm}^3 neither rises nor sinks but hangs in the water.

The units must belong together. Grams go with cubic centimeters and kilograms with cubic meters. A volume in cubic meters divided into a mass in grams gives a true rate in an unfamiliar unit, not a number to compare with \text{g/cm}^3.

Population density is an average. A district with 4{,}000 people per square kilometer may have crowded streets and an empty park; the number says nothing about any one square kilometer.

Where this goes next

Density is one of the Rates from measurements, numbers made by dividing one measurement by another. Together with Flow rate and speed it shows one idea, an amount per one unit, measuring very different things. Modeling with solids: density and design uses density to find the mass of cylinders, cones and spheres, where the volume takes more work but the step m = D \times V is the same. Population density turns up again in estimates such as how many people live on a whole continent.

Exercises

Finding density

  1. A slate sample has mass 56 g and volume 20\ \text{cm}^3. Find its density.
  2. A block of ice 10 cm by 10 cm by 5 cm has mass 460 g. Find its density. Does ice float?
  3. A copper cube has edges of 2 cm and mass 72 g. Find the density of copper.
  4. A brick 20 cm by 10 cm by 6 cm has mass 2{,}400 g. Find its density.

Finding mass or volume

  1. Olive oil has a density of about 0.92\ \text{g/cm}^3. Find the mass of the oil in a full 750\ \text{cm}^3 bottle.
  2. Gold has a density of about 19.3\ \text{g/cm}^3. A gold nugget has mass 38.6 g. Find its volume.
  3. Iron has a density of about 7.9\ \text{g/cm}^3. What volume of iron has a mass of 316 g?
  4. A fish tank 40 cm by 25 cm by 30 cm is full of water. Find the mass of the water in grams and in kilograms.
  5. Concrete has a density of about 2{,}400\ \text{kg/m}^3. Find the mass of a concrete slab 3 m by 2 m by \tfrac{1}{10} m.
  6. Cork has a density of about 0.24\ \text{g/cm}^3. Find the mass of a cork stopper with volume 15\ \text{cm}^3.

Comparing and floating

  1. Use Figure 2. Which of cork, pine, glass, granite and iron float in water?
  2. Balsa wood has a density of about 160\ \text{kg/m}^3. Write this in \text{g/cm}^3 and say whether balsa floats.
  3. Rock sample A has mass 120 g and volume 40\ \text{cm}^3. Rock sample B has mass 200 g and volume 80\ \text{cm}^3. Which has more mass? Which is denser?
  4. Two cubes each have edges of 3 cm. One has mass 67.5 g and the other 10.8 g. Find each density and say which cube floats.

Population density

  1. A town has 18{,}000 people on 6\ \text{km}^2. Find its population density.
  2. Valley A has 2{,}400 people on 40\ \text{km}^2. Valley B has 1{,}500 people on 30\ \text{km}^2. Which valley is more crowded?
  3. A city has 5{,}000 people per square kilometer and covers 9\ \text{km}^2. How many people live there?
  4. A village has 600 people and a population density of 150 people per square kilometer. What is its area?

Reasoning

  1. A large block of foam has more mass than a steel coin. Explain how the foam can still be less dense.
  2. A rock sample is cut in half. What happens to its mass, its volume and its density? Explain.
  3. A piece of pumice floats when it is first dropped in a pond. Weeks later, after water has slowly seeped into its tiny air bubbles, it sinks. Its outside size has not changed. Explain using density.
  4. Two towns have the same number of people, and one has twice the area of the other. How do their population densities compare? Explain.
Answers
  1. 2.8\ \text{g/cm}^3. 56 \div 20 = 2.8.
  2. 0.92\ \text{g/cm}^3, so ice floats. V = 10 \times 10 \times 5 = 500\ \text{cm}^3 and 460 \div 500 = 0.92, less than 1.
  3. 9\ \text{g/cm}^3. V = 2 \times 2 \times 2 = 8\ \text{cm}^3 and 72 \div 8 = 9.
  4. 2\ \text{g/cm}^3. V = 20 \times 10 \times 6 = 1{,}200\ \text{cm}^3 and 2{,}400 \div 1{,}200 = 2.
  5. 690 g. m = 0.92 \times 750 = 690.
  6. 2\ \text{cm}^3. V = 38.6 \div 19.3 = 2.
  7. 40\ \text{cm}^3. V = 316 \div 7.9 = 3160 \div 79 = 40.
  8. 30{,}000 g, which is 30 kg. V = 40 \times 25 \times 30 = 30{,}000\ \text{cm}^3, and each cubic centimeter of water has mass 1 g.
  9. 1{,}440 kg. V = 3 \times 2 \times \tfrac{1}{10} = \tfrac{6}{10}\ \text{m}^3 and 2{,}400 \times \tfrac{6}{10} = 1{,}440.
  10. 3.6 g. m = 0.24 \times 15 = 3.6.
  11. Cork and pine float; glass, granite and iron sink. Only cork (0.24) and pine (0.5) are below 1\ \text{g/cm}^3.
  12. 0.16\ \text{g/cm}^3, so balsa floats. 160 \div 1{,}000 = 0.16, less than 1.
  13. B has more mass, but A is denser. A: 120 \div 40 = 3\ \text{g/cm}^3; B: 200 \div 80 = 2.5\ \text{g/cm}^3.
  14. 2.5\ \text{g/cm}^3 and 0.4\ \text{g/cm}^3; the second cube floats. Each has volume 3 \times 3 \times 3 = 27\ \text{cm}^3, and 67.5 \div 27 = 2.5, 10.8 \div 27 = 0.4.
  15. 3{,}000 people per square kilometer. 18{,}000 \div 6 = 3{,}000.
  16. Valley A is more crowded. Valley A: 2{,}400 \div 40 = 60 people per square kilometer against 1{,}500 \div 30 = 50 in Valley B.
  17. 45{,}000 people. 5{,}000 \times 9 = 45{,}000.
  18. 4\ \text{km}^2. 600 \div 150 = 4.
  19. Mass is about the whole object; density is about one cubic centimeter. The foam has a great many cubic centimeters, each with very little mass, so its total can beat the coin’s while each of its cubic centimeters holds far less than one of steel.
  20. Mass and volume are both halved, and the density stays the same: each half is the same rock, and \tfrac{1}{2}m \div \tfrac{1}{2}V = m \div V.
  21. The pumice’s volume stays the same, but its mass grows: water, with far more mass per cubic centimeter than air, has taken the air’s place. Mass divided by the same volume is now larger. Once that density passes 1\ \text{g/cm}^3, the piece is denser than water and sinks.
  22. The larger town’s density is half the other’s. The same people spread over twice the area gives P \div 2A = \tfrac{1}{2}(P \div A).
kg1128 · updated 2026-10-13 · CC-BY-SA-4.0