5th Grade · 5.MD.C.3 · 5.MD.C.4

What is volume in 5th grade math?

What is volume? In 5th grade math, volume is the amount of space inside a solid shape, such as a box, not how loud a sound is. Your child learns to measure volume by filling a shape with small cubes and counting them, and writes the answer in cubic units.

What the standards say

The official Common Core wording, quoted exactly. The plain version follows.

5.MD.C.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.C.3.a A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
5.MD.C.3.b A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

5.MD.C.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

What it means, in plain words

Volume and the unit cube 5.MD.C.3

A solid shape, also called a 3D shape, has length, width and height, like a box or a brick. A flat shape drawn on paper, like a square, has no volume because it has no thickness. Volume tells how much space a solid shape takes up, or how much it can hold.

To measure volume, your child uses a unit cube: a cube whose edges are all 1 unit long. One unit cube has a volume of 1 cubic unit. If a solid can be packed with 20 unit cubes, with no gaps and no overlaps, its volume is 20 cubic units. A simple answer to "what is volume" is: how many unit cubes fit inside.

Volume is different from area. Area counts the flat squares that cover a surface and is measured in square units. Volume counts the cubes that fill a space and is measured in cubic units.

Your child should be able to:

Explain volume as the space inside a solid shape
Tell volume apart from length and area
Describe a unit cube and one cubic unit
Find a volume by packing a shape with unit cubes, with no gaps or overlaps

Example

One shape is a tower of 6 unit cubes. Another is a flat row of 6 unit cubes. Which shape has the greater volume?

Each unit cube has a volume of 1 cubic unit.
The tower is made of 6 unit cubes, so its volume is 6 cubic units.
The row is also made of 6 unit cubes, so its volume is also 6 cubic units.
The two shapes have the same volume. Moving the cubes around changes the shape, not the volume.

Counting cubes to measure volume 5.MD.C.4

Counting cubes volume problems show a solid built from cubes, and your child counts how many there are. The cubes can have real sizes. A cube with 1-centimeter edges is 1 cubic centimeter, and a cube with 1-inch edges is 1 cubic inch. Bigger things, like a closet, are measured in cubic feet. Your child also measures with improvised units, any small objects of the same size, such as sugar cubes.

Counting one cube at a time is slow, and some cubes are hidden at the back or underneath. So your child learns to count in layers: count the cubes in the bottom layer, count the layers, then multiply. Volume with unit cubes is the base for the formula that comes next, length times width times height.

Your child should be able to:

Count unit cubes to find the volume of a solid, including cubes that are hidden
Count in layers: the cubes in one layer times the number of layers
Give the answer in cubic centimeters, cubic inches or cubic feet
Measure the volume of a box with improvised units, such as sugar cubes

Example

A box is filled with 1-centimeter cubes. The bottom layer has 3 rows of 4 cubes, and there are 3 layers. What is the volume of the box?

One layer: $3 \times 4 = 12$ cubes.
Three layers: $12 + 12 + 12 = 36$ , which is the same as $12 \times 3 = 36$ .
Each cube is 1 cubic centimeter, so the volume of the box is 36 cubic centimeters.

Where children usually get stuck

Your child counts only the cubes that show in the picture, and the answer comes out too small.
A drawing shows the front, the top and one side. The cubes in the middle and at the back are there, but nothing on the page shows them.
What helps. Build a small box from blocks or sugar cubes. Count it by layers, then lift off the top layer so your child can see the cubes that were hidden.
Your child counts the small squares on the outside faces of the box instead of the cubes.
Area lessons taught your child to count squares, and the faces of the cubes look like squares. The same cube can show up on two or three faces, so it gets counted more than once.
What helps. Point to one corner cube and count how many of its faces show. Then ask whether it is one cube or three.
Your child writes the answer as 36 square centimeters, or just 36.
Area answers end in square units, and the habit carries over. The unit says what was counted, and here the count was of cubes.
What helps. Ask what was counted, squares or cubes. If it was cubes, the answer is in cubic units.

Try four practice questions

Real 5th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up

The table below shows 4 statements about what volume measures.

Number	Statement
1	Volume measures how much space is inside a solid figure.
2	Volume measures how many faces a solid figure has.
3	Volume measures how much of the outside of a solid is covered.
4	Volume measures the total length of all the edges on a solid.

Mark each statement True or False in the response table.

Show the answer and how to solve it

Statement 1 is true. Statements 2, 3, and 4 are false because volume measures the space inside a solid figure, not its faces, outside, or edge lengths.

1Read the statement table. Volume measures how much space is inside a solid figure.
2Statement 1 is true because it matches the definition of volume exactly.
3Statement 2 is false. Counting faces tells about parts of a solid, not the space inside it.
4Statement 3 is false. The outside of a solid being covered is not what volume measures.
5Statement 4 is false. The total edge length of a solid tells about its edges, not about inside space.

Common mistake. Adding all edge lengths gives a length, not a volume. Edge lengths can help calculate volume, but volume measures the space inside a solid.

The idea underneath. Volume measures the amount of three-dimensional space inside a solid figure.

2On level
The two views below show the same right rectangular prism built from 1-inch cubes. Its volume is measured in cubic inches.
Front View
Back View
Mia counted the cubes. Mia's work is shown below.
$\displaystyle 4 + 4 + 4 = 12\text{ cubic inches}$
Select the statement that best describes Mia's work.
A
Mia's work misses or double-counts a whole row or group of cubes.
B
Mia's work is correct.
C
Mia's work counts only one face of the prism.
D
Mia's work adds edge lengths instead of counting cubes.
Show the answer and how to solve it
Mia's work is correct because the prism has 3 layers of 4 cubes each, and Mia counted all 3 groups to get 12 cubic inches.
1. 1Use the views to organize the prism into horizontal layers. Each layer is $2 \times 2 = 4$ cubes.
2. 2There are 3 equal layers, so the correct cube count adds 3 groups of 4.
3. 3Now compare that count to Mia's work.
4. 4Mia's expression has exactly 3 groups of 4, one for each layer, and the total of 12 cubic inches is correct.
5. 5The best description is that Mia's work is correct.
Common mistake. A common mistake is assuming an error exists when the work is actually right. Verify the group size $(2 \times 2 = 4)$ and the number of groups (3 layers) both match the prism before deciding work is wrong.
The idea underneath. Counting volume with unit cubes means counting every cube exactly once; equal groups help organize the count.
3On level
The solid figure below helps show what volume means.
Select a choice from each dropdown to complete the sentences.
Volume measures
. It is
the solid figure.
Show the answer and how to solve it
The correct choices are 'space' and 'inside' because volume describes how much space is inside a solid figure.
1. 1The picture shows a solid figure, so it has space on the inside.
2. 2Volume tells how much space a solid figure takes up. It is not about area, edge length, or faces.
3. 3The completed sentences are: Volume measures space. It is inside the solid figure.
Common mistake. A common mistake is choosing 'area' for the first blank, which only describes flat, two-dimensional figures. Volume is about the space a solid figure takes up inside.
The idea underneath. Volume is a three-dimensional attribute: it describes the amount of space a solid figure takes up, not just its outside boundary.
4Stretch
The solid below is made of 1-inch cubes. Find its volume, including the hidden cubes.
Enter the volume.
Volume:cubic inches
Show the answer and how to solve it
Answer. 16
The volume is 16 cubic inches because the 7 column heights add to 16.
1. 1Split the solid into 2 rows of columns. In this view, the front row is blue and the back row is gold.
2. 2The front row has 3 columns with heights 1, 2, and 1.
3. 3The back row has 4 columns with heights 3, 4, 3, and 2. The first 3 back columns have hidden lower cubes. The far-right back column has no front column in front of it.
4. 4Add the two row totals to find the volume.
Common mistake. A common mistake is counting only the cubes visible above the front row. Count every back column at its full height, including hidden cubes.
The idea underneath. An irregular cube stack can be counted by treating each floor spot as one vertical column and adding the column heights.