Grade 5 · 5.MD.C.3 · 5.MD.C.3A

Volume as filling

A building block, a jug of water and an oddly shaped stone all take up space. How do you say how much, and how do you compare them?

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Liquid volume: liters and millilitersGrade 3

Where you are 9 in Volume and capacity

Measurement12

Volume and capacity9

▴ 1 above Liquid volume: liters and milliliters Customary capacity units

Volume as filling

Volume with unit cubes Volume of a box ▾ Show all 9 (3 more)

Why this exists

Length says how long something is. Area says how much flat space a shape covers, and you find it by covering the shape with unit squares and counting them. Neither one tells you how much room a solid thing takes up. A tall thin tower of blocks and a low wide wall of blocks can use exactly the same blocks, so their heights and their floor areas say nothing about which is “bigger”.

Volume answers that question. It measures space in three directions at once, and it is found the same way area is: choose a small piece of space as the unit, then count how many copies of it fill the solid. The piece is a cube.

The same idea covers two other everyday questions. How much does a jug or a toy chest hold? That is the volume of the empty space inside it. And how much space does a stone take up, when its bumpy shape could never be filled with cubes? Sinking it in water turns that question into a reading on a measuring jug. This lesson takes the three in turn, one part each.

Definition

A unit cube is a cube whose edges are each 1 unit long. Its volume is 1 cubic unit. A cube 1 centimeter on each edge is a cubic centimeter; a cube 1 inch on each edge is a cubic inch.

The volume of a solid is the number of unit cubes that fill it exactly, with no gaps and no overlaps. A solid that 10 unit cubes fill has a volume of 10 cubic units.

The capacity of a container is the volume of the space inside it: how much it can hold. Liquids are measured in liters and milliliters:

1 \text{ L} = 1000 \text{ mL}

Intuition

Part 1: filling a solid with unit cubes

Think of a building block shaped like a cube as one unit of space. Kiona’s blocks are all the same size, so any solid she builds from them has a volume she can count: one cubic unit per block. In the first figure, solid A and solid B look different, but each uses 6 blocks, so each has a volume of 6 cubic units.

Figure 1A unit cube, and two solids built from the same 6 blocks. Their shapes differ; their volumes do not.

The two conditions in the definition matter. No gaps: if the cubes leave empty spaces, the count is too small for the space being measured. No overlaps: two cubes cannot sit in the same place, so no piece of space is counted twice. Cubes are chosen as the unit because cubes stack face to face and leave no gaps at all. Balls or cylinders would always leave gaps between them.

A drawing of a block solid shows only the faces turned toward you. Blocks behind or underneath others may be completely hidden. In a solid built from blocks, every block rests on the floor or on another block, so when you see the top of a stack, the whole stack is under it. Counting stack by stack is a reliable way to find the blocks you cannot see.

Part 2: volume and capacity

A solid takes up space; a container holds space. A toy chest takes up room on the floor, and it holds toys. These are two different volumes. The space the chest takes up includes its wooden walls; the space it holds does not. So a container always takes up more space than it can hold, and thick walls make the difference bigger.

Liquids have no shape of their own, so they are not counted in blocks. They are measured by pouring them into a container marked in milliliters or liters. A milliliter is a small amount: a teaspoon holds about 5 mL. A liter is 1000 of them, the amount in a large water bottle. A milliliter and a cubic centimeter are the same amount of space, which Volume with unit cubes shows.

Pouring changes the shape of the water but not how much water there is. That idea has its own star, Volume does not change when it is poured.

Part 3: measuring an odd shape with water

A stone has no flat faces, so no number of cubes fills it exactly. Water does the filling instead. When the stone sinks into a jug of water, it pushes water out of the space it now occupies, and the only place that water can go is up. The water level rises.

Figure 2Sinking a stone lifts the reading from 400 mL to 550 mL. The stone takes up the space of 150 mL of water.

The amount of water has not changed. So the rise in the reading is exactly the space the stone takes up. This way of finding volume is called displacement: the stone displaces, or moves aside, its own volume of water. Madhav sees the same thing when he climbs into a bathtub and the water rises up its sides.

Results

Volume by counting. A solid that is filled exactly, with no gaps and no overlaps, by n unit cubes has a volume of n cubic units.

Volumes add. Put two solids together without overlapping, and the volume of the new solid is the sum of their volumes. Take a part away, and you subtract its volume.

Rearranging keeps the volume. Solids built from the same unit cubes have the same volume, whatever their shapes.

Capacity. The capacity of a container is the volume of the empty space inside it. The space a container takes up is its capacity plus the volume of its walls.

Liters and milliliters. 1 \text{ L} = 1000 \text{ mL}.

Displacement. When a solid sinks completely under water,

\begin{aligned} \text{volume of the solid} &= \text{reading after} \\ &\quad - \text{reading before} \end{aligned}

Why it is true

Volumes add. If solid A is filled by a unit cubes and solid B by b unit cubes, then the solid made by joining them is filled by those same cubes together, a + b of them. None of them overlap, because A and B do not overlap, and no gap opens up, because each part was already full. So the joined solid has volume a + b. Rearranging is the same argument: moving cubes to new places does not change how many there are.

Capacity. Fill a container to the top. The space inside is filled exactly by its contents, so the contents have the same volume as that space. The container itself takes up the inside space plus the space of its walls, and by “volumes add” that is the capacity plus the volume of the walls.

Displacement. Water can be poured into any shape, but it cannot be squeezed into a smaller amount of space. Before the stone goes in, the space below the water line holds only water, and the reading tells you how much. After the stone sinks, the space below the new water line holds the same water plus the stone. So

\begin{aligned} \text{reading after} &= \text{water} + \text{stone} \\ \text{reading before} &= \text{water} \end{aligned}

and subtracting leaves the stone alone. The shape of the jug does not matter, because its marks already say how much space sits below each line.

Worked examples

Examples 1 to 3 go with Part 1, Examples 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Example 1

Same blocks, two solids

Kiona builds solid A in the first figure: a row of 4 blocks on the floor with 2 more stacked on the left end. Then she takes it apart and builds solid B: a wall 3 blocks wide and 2 blocks high. Which solid has the greater volume?

Solution

Count the unit cubes in each. Solid A has 4 blocks in the row and 2 on top, so it has 4 + 2 = 6 blocks and a volume of 6 cubic units. Solid B has 3 blocks in the bottom row and 3 in the top row, so 3 + 3 = 6 blocks, also 6 cubic units.

The volumes are equal. That had to happen: B was built from the very same blocks as A, and moving blocks around does not change how many there are. Solid A is taller and B is wider, but volume measures space, not height or width.

Example 2

Counting the blocks you cannot see

Madhav builds the staircase drawn below. It is 2 blocks deep, and from left to right its steps are 3, 2 and 1 blocks high. What is its volume? How many blocks cannot be seen at all in the drawing?

Solution

Figure 3The staircase in Example 2. Some blocks in the back row cannot be seen at all, but they must be there.

Count stack by stack. Each step is two stacks, one in front and one behind, and each stack stands on the floor, so a stack of a given height holds that many blocks. The front row has stacks of 3, 2 and 1, which is 3 + 2 + 1 = 6 blocks. The back row has the same stacks, 6 more. The volume is

6 + 6 = 12 \text{ cubic units}

Now the hidden ones. All 6 front blocks show their front faces. In the back row, only the top block of each stack shows: its top and its right side. That is 3 back blocks that show. The other 6 - 3 = 3 back blocks are covered on every side you can see: the two lower blocks behind the tallest step, and the bottom block behind the middle step. So 3 blocks are completely hidden, and counting only what shows would give 9, which is too few.

Example 3

Two sizes of block in a toy chest

Xuan fills a toy chest with big cube-shaped blocks, packed with no gaps, and it takes exactly 24 of them. Her small blocks are cubes too: 8 small blocks fit together exactly into one big block. How many small blocks would fill the chest? What is the volume of the chest?

Solution

Each big block in the chest could be swapped for the 8 small blocks that make it, leaving no gaps and no overlaps. So the chest holds 8 small blocks for each of its 24 big ones:

24 \times 8 = 192 \text{ small blocks}

Both counts describe the same space. The volume of the chest is 24 cubic units when the unit is a big block, and 192 cubic units when the unit is a small block. A volume is a number together with its unit, so the unit must always be named. This is why people agree on standard units such as the cubic centimeter: everyone’s centimeter cube is the same size.

Example 4

What a container holds and what it takes up

Madhav builds the open box drawn below. Its floor is a layer of 9 blocks, 3 by 3. On top he puts a ring of 8 blocks around the edge, leaving the middle of the top layer empty. How many blocks did he use? How much space does the box take up on the shelf? What is its capacity, in blocks?

Solution

Figure 4The open box in Example 4. It is made of 17 blocks, takes up the space of 18, and holds 1.

He used 9 blocks in the floor and 8 in the ring, so

9 + 8 = 17 \text{ blocks}

That is the volume of the walls and floor, the material of the box.

On the shelf, the box takes up a space that would be filled by 9 blocks below and 9 above if the middle were filled in, so it takes up 9 + 9 = 18 cubic units. The empty middle holds exactly one block, so the capacity is 1 cubic unit.

The three numbers fit the Results: the space taken up is the capacity plus the walls, 1 + 17 = 18. A box with thick walls like this one can take up a lot of space and hold very little.

Example 5

How much more fits in the jug

Xuan’s measuring jug has a capacity of 2 L. She pours in 750 mL of water, then 600 mL more. How much water is in the jug, and how much more will fit before it is full?

Solution

Change the capacity to milliliters so that every amount uses the same unit. Since 1 \text{ L} = 1000 \text{ mL}, the jug holds 2 \times 1000 = 2000 mL.

The water in the jug is 750 + 600 = 1350 mL, which is 1 L 350 mL. The space still empty is

2000 - 1350 = 650 \text{ mL}

so 650 mL more will fit. The capacity is still 2 L, whatever amount is in the jug: capacity belongs to the container, not to what happens to be inside it.

Example 6

A stone in a measuring jug

Kiona’s measuring jug holds water up to the 400 mL mark. She lowers a stone in, and it sinks to the bottom. The water rises to the 550 mL mark, as in the jug figure in Part 3. What is the volume of the stone?

Solution

The water is the same 400 mL as before; nothing was added or poured out. After the stone goes in, the space below the water line holds that water and the stone together, 550 mL of space. So the stone takes up

550 - 400 = 150 \text{ mL}

The volume of the stone is 150 mL, the same space as 150 cubes 1 centimeter on each edge. The stone’s shape never had to be filled with cubes; the water filled around it.

Example 7

Marbles one at a time

Madhav’s jug has a capacity of 1 L and holds 500 mL of water. He drops in 8 identical marbles, and the reading rises to 540 mL. What is the volume of one marble? How many marbles must be in the jug, in all, for the reading to reach 600 mL?

Solution

The 8 marbles together push the reading up by 540 - 500 = 40 mL, so together they have a volume of 40 mL. They are identical, so they share that space equally:

40 \div 8 = 5 \text{ mL each}

To reach 600 mL, the marbles must take up 600 - 500 = 100 mL of space in all. Each takes up 5 mL, so that needs 100 \div 5 = 20 marbles, which is 20 - 8 = 12 more than are in the jug now.

Check. 20 marbles take up 20 \times 5 = 100 mL, and 500 + 100 = 600 mL. That is below the 1000 mL capacity, so no water spills and the reading can be trusted.

Measuring all 8 at once was a good choice. On a jug marked every 10 mL, one marble alone would lift the water only 5 mL, half a mark, which is hard to read; eight lift it 40 mL, four whole marks.

Boundaries

Counting cubes needs an exact fit. The count is the volume only when the cubes fill the solid with no gaps and no overlaps. A solid with curved or slanted faces, such as a ball or a stone, cannot be filled exactly by whole cubes, so counting cubes only gives an estimate there. A volume found by counting is in the unit of the cube used, and a different cube gives a different number.

Volume is not height or area. A taller solid need not have a greater volume, and neither need a solid with a larger floor. Only the count of cubes, or another full measure of space, decides.

A drawing may hide blocks. The count from a picture is right only if you can tell where every block is. A solid of resting blocks has no holes under a top you can see; a solid that might have hidden holes, or blocks glued so they hang in the air, cannot be counted from one picture.

Capacity counts only the inside. The capacity of a container is never the space it takes up, unless its walls had no thickness at all.

Displacement needs the right object and the right jug. The object must be completely under the water; if it floats, push it under with something as thin as a wire, or part of it is left out of the reading. It must not soak up water (a sponge) or dissolve in it (sugar), because then the water itself changes. The water must not overflow before the reading is taken, unless the spilled water is caught and measured, and the jug’s marks must be fine enough to read the rise.

Where this goes next

Counting one block at a time is slow for a big box. Volume with unit cubes packs a box in whole layers of cubes and counts a layer at a time, and Volume of a box turns that into a rule using the box’s three edge lengths. The same filling idea still works when the edges are fractions, in Volume of a box with fraction edge lengths, and for solids made of several boxes, in Volume of a solid made of boxes.

Capacity and displacement come back in science. Once a stone’s volume is known by displacement and its mass is weighed, the two together give its density.

Exercises

Part 1: counting cubes

Kiona lays a row of 5 blocks on the floor, then stands 3 more blocks on top of the row, each on its own block. What is the volume of the solid?
Madhav stacks 7 blocks into a tower, and Xuan lays 7 blocks in a row on the floor. Which solid has the greater volume? What is each volume?
A staircase is 1 block deep, and from left to right its steps are 4, 3, 2 and 1 blocks high. What is its volume? What would the volume be if it were built 2 blocks deep, with the same steps?
A plan of a block solid seen from above shows the height of each stack, in blocks:

3	1	2
2	2	1

What is the volume of the solid? 5. Another plan shows these stack heights:

1	2	1
2	3	2
1	2	1

What is the volume? Looking straight down from above, how many of the blocks cannot be seen? 6. A toy chest is filled exactly by 30 big blocks. Each big block can be built exactly from 8 small blocks. How many small blocks fill the chest? 7. Kiona’s solid has 7 blocks and Xuan’s has 9. They join the two without overlapping. What is the volume of the new solid? Madhav then takes 4 blocks off it. What is the volume now?

Part 2: volume and capacity

Xuan builds an open box. The bottom layer has 12 blocks. On it she puts a layer of 10 blocks around the edge, leaving an empty gap the size of 2 blocks. How many blocks did she use? What is the capacity of the box, in blocks? How much space does it take up?
A jug with a capacity of 1 L holds 420 mL of water. How much more water fits?
How many cups of 250 mL each can be filled from a full 2 L bottle?
Madhav pours 3 L of water into glasses that hold 400 mL each. How many glasses can he fill completely, and how much water is left over?
Write 1 L 250 mL and 3 L 75 mL in milliliters. Write 4{,}500 mL in liters and milliliters.
A jug has a capacity of 1{,}500 mL. Kiona pours in 3 cups of water with 300 mL in each. How much water is in the jug, and how much more fits?

Part 3: displacement

A jug reads 200 mL. After a stone sinks in it, the jug reads 275 mL. What is the volume of the stone?
A jug reads 650 mL. Xuan sinks a metal toy in it, and the reading becomes 890 mL. What is the volume of the toy?
Madhav drops 5 identical marbles into a jug that reads 300 mL, and the reading rises to 330 mL. What is the volume of one marble?
A stone with a volume of 120 mL sinks into a jug holding 450 mL of water. The capacity of the jug is 600 mL. What does the jug read now? A second stone just like the first is added. How much water spills over?
A bowl is filled with water right to the brim and set in a tray. Kiona lowers a metal toy into it until the toy is completely under water. She pours the water that spilled into the tray into a measuring cup that already held 50 mL, and the cup now reads 230 mL. What is the volume of the toy?
A jug reads 400 mL. Xuan sinks a toy car in it, and it reads 460 mL. Then she sinks a stone in as well, and it reads 545 mL. What are the volumes of the car and of the stone? What do they take up together?

Reasoning

Why is volume measured by filling with cubes rather than with balls?
Madhav says his tower of 6 blocks has a greater volume than Kiona’s flat layer of 6 blocks, because it is taller. Is he right? Explain.
Why would sinking a dry sponge in a jug of water give the wrong volume for the sponge? What has to be done with a cork, which floats?
Explain why the rise in the water reading equals the volume of the stone, and why the shape of the jug does not change the answer.
A box has thick wooden walls. Which is greater, the space the box takes up or its capacity? Explain.
Xuan counts 10 blocks she can see in a drawing of a block solid and says its volume is 10 cubic units. Explain why the volume could be more than 10, and why it cannot be less.

Answers

8 cubic units. The row has 5 blocks and 3 stand on it: 5 + 3 = 8.
Neither; each has a volume of 7 cubic units. Both use the same number of blocks.
10 cubic units, then 20 cubic units. One deep: 4 + 3 + 2 + 1 = 10. Two deep, the back row repeats the front row: 10 + 10 = 20.
11 cubic units. Adding the stacks: 3 + 1 + 2 + 2 + 2 + 1 = 11.
15 cubic units, and 6 blocks cannot be seen. The stacks add to 1 + 2 + 1 + 2 + 3 + 2 + 1 + 2 + 1 = 15. From above you see only the top block of each of the 9 stacks, so 15 - 9 = 6 are hidden.
240 small blocks. Each of the 30 big blocks is 8 small ones: 30 \times 8 = 240.
16 cubic units, then 12 cubic units. Volumes add: 7 + 9 = 16; taking blocks away subtracts: 16 - 4 = 12.
22 blocks used; capacity 2 cubic units; it takes up 24 cubic units. Blocks: 12 + 10 = 22. The gap holds 2 blocks. The space taken up is the blocks plus the gap, 22 + 2 = 24.
580 mL. The jug holds 1000 mL, and 1000 - 420 = 580.
8 cups. The bottle holds 2000 mL, and 2000 \div 250 = 8.
7 glasses, with 200 mL left over. 3 L is 3000 mL; 7 \times 400 = 2800, and 3000 - 2800 = 200 mL, too little for an eighth glass.
1250 mL; 3075 mL; 4 L 500 mL. 1000 + 250 = 1250; 3000 + 75 = 3075; and 4{,}500 = 4000 + 500, which is 4 L and 500 mL.
900 mL in the jug, and 600 mL more fits. 3 \times 300 = 900, and 1500 - 900 = 600.
75 mL. The rise is 275 - 200 = 75.
240 mL. The rise is 890 - 650 = 240.
6 mL. The 5 marbles take up 330 - 300 = 30 mL, and 30 \div 5 = 6.
It reads 570 mL; then 90 mL spills. First, 450 + 120 = 570. The second stone would need room for 570 + 120 = 690 mL, but the jug holds only 600, so 690 - 600 = 90 mL of water spills over.
180 mL. The spilled water is 230 - 50 = 180 mL. The bowl was full, so every bit of space the toy took had to push the same amount of water over the brim.
Car 60 mL, stone 85 mL, together 145 mL. Car: 460 - 400 = 60. Stone: 545 - 460 = 85. Together: 545 - 400 = 145, which matches 60 + 85 = 145.
Cubes fit face to face and fill space with no gaps. Balls touch only at points and always leave gaps between them, so counting balls would leave some of the space unmeasured.
No. Volume is the number of unit cubes that fill a solid, and both solids use 6 blocks, so both have a volume of 6 cubic units. Being taller changes the shape, not the amount of space.
A sponge is full of holes that soak up water. Water moves into the sponge instead of rising, so the rise is too small. A cork floats, so part of it sits above the water and is not measured; it must be pushed fully under with something very thin, such as a wire.
The water is not added to or poured out, and it cannot be squeezed smaller, so the amount of water stays the same. After the stone sinks, the space below the water line holds that water plus the stone. The reading after minus the reading before leaves just the stone. The jug’s marks already tell how much space lies below each line, whatever the jug’s shape.
The space it takes up. It takes up its capacity plus the space of its walls, and thick walls take up a lot of space.
Blocks behind or under others may be hidden, so there may be more than 10. It cannot be fewer, because each of the 10 blocks she sees is really there, and two blocks cannot share the same space.