5th Grade · 6th Grade · 5.MD.C.5 · 6.G.A.2

Volume of rectangular prisms in 5th and 6th grade

The volume of rectangular prisms is the space inside a box shape, measured in cubic units. In 5th grade, your child learns the formulas $V = l \times w \times h$ and $V = b \times h$ for boxes with whole-number edge lengths. In 6th grade, the same formulas work with fractions, such as a box that is $\frac{1}{2}$ foot tall.

What the standards say

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

5.MD.C.5 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.C.5.a Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
5.MD.C.5.b Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
5.MD.C.5.c Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

6.G.A.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

What it means, in plain words

5th grade: from counting cubes to the formula 5.MD.C.5

A rectangular prism is a solid shape with 6 rectangle faces, like a cereal box or a brick. Its edges have three lengths: the length ( $l$ ), the width ( $w$ ) and the height ( $h$ ). The volume of rectangular prisms in 5th grade uses whole-number lengths.

Your child first packs a prism with unit cubes and sees a pattern. The bottom layer holds $l \times w$ cubes, and there are $h$ layers. So the volume of a rectangular prism is $V = l \times w \times h$ . Another way to say it: the area of the base, $b$ , times the height, so $V = b \times h$ . Both formulas give the same answer.

Volume can also be added. If a solid is made of two boxes that do not overlap, your child finds the volume of each box and adds the two. Volume word problems in 5th grade use this for shapes like an L-shaped room or two stacked blocks.

Your child should be able to:

Find the volume of a rectangular prism by multiplying the length, the width and the height
Use $V = b \times h$ when the area of the base is given
Find the volume of a solid made of two rectangular prisms by adding their volumes
Solve volume word problems and give the answer in cubic units

Example

A shoebox is 12 inches long, 6 inches wide and 5 inches tall. What is its volume?

$V = l \times w \times h = 12 \times 6 \times 5$
$12 \times 6 = 72$ , and $72 \times 5 = 360$ .
The shoebox has a volume of 360 cubic inches.

Example

A solid is made of two rectangular prisms that do not overlap. The bottom one is 8 cm long, 3 cm wide and 2 cm tall. The top one is 3 cm long, 3 cm wide and 4 cm tall. What is the total volume?

Bottom prism: $8 \times 3 \times 2 = 48$ cubic centimeters.
Top prism: $3 \times 3 \times 4 = 36$ cubic centimeters.
Add the two: $48 + 36 = 84$ .
The solid has a volume of 84 cubic centimeters.

6th grade: edge lengths that are fractions 6.G.A.2

The volume of rectangular prisms in 6th grade uses edges that can be fractions, such as $\frac{3}{4}$ inch or $2\frac{1}{2}$ feet. The formulas do not change. Your child still multiplies the length, the width and the height, or the area of the base by the height. The new part is multiplying fractions and mixed numbers to do it.

To see why the formula still works, your child packs the prism with small cubes whose edges are a unit fraction, such as $\frac{1}{2}$ unit. A unit fraction is a fraction with 1 on top. Eight of these small cubes fit in one unit cube: 2 across, 2 deep and 2 high. So each small cube is $\frac{1}{8}$ of a cubic unit. Counting the small cubes and multiplying by $\frac{1}{8}$ gives the same answer as the formula.

Your child should be able to:

Multiply three fractions or mixed numbers to find a volume
Explain why a cube with $\frac{1}{2}$ -unit edges has a volume of $\frac{1}{8}$ cubic unit
Use $V = l \times w \times h$ or $V = b \times h$ with fraction edge lengths
Solve word problems about boxes and containers with fraction measurements

Example

A sandbox is 4 feet long, $2\frac{1}{2}$ feet wide and $\frac{1}{2}$ foot deep. How much sand does it take to fill it?

Write the mixed number as a fraction: $2\frac{1}{2} = \frac{5}{2}$ .
$V = 4 \times \frac{5}{2} \times \frac{1}{2} = \frac{4 \times 5 \times 1}{2 \times 2} = \frac{20}{4}$
$\frac{20}{4} = 5$
It takes 5 cubic feet of sand to fill the sandbox.

Where children usually get stuck

Your child finds the shoebox volume and writes 360 square inches.
Area problems end in square units, and the habit carries over. Volume counts cubes, so the unit is cubic inches.
What helps. Ask what the 360 counts: flat squares or cubes that fill the box? Cubes means cubic units.
For the solid made of two prisms, your child uses the height of the whole solid, 6 cm, for the top prism.
The picture labels the whole solid, and some labels belong to only one of the two boxes. The child does not check which lengths belong to which box.
What helps. Color the two boxes differently on the picture. Then list the three lengths of each box on its own before multiplying.
Your child works out $2\frac{1}{2} \times 4$ as $2 \times 4$ with the $\frac{1}{2}$ tacked on, and gets $8\frac{1}{2}$ .
Only the whole number was multiplied. The $\frac{1}{2}$ has to be multiplied by 4 as well: $8 + 2 = 10$ .
What helps. Change the mixed number into a fraction first, $\frac{5}{2}$ . Or say it as groups: 2 groups of 4 is 8, and half a group of 4 is 2 more.

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
A right rectangular prism is shown in the diagram below. All edge lengths are in centimeters.
Prism A
Fill in the prism's edge lengths to complete the volume equation. Do not multiply.
$\displaystyle V_A =$
$\displaystyle \times$
$\displaystyle \times$
Show the answer and how to solve it
$V_A = 5 \times 7 \times 2$ . The order of the 3 factors can change, but the equation must use all 3 edge lengths from Prism A.
1. 1Look at Prism A. Its 3 edge lengths are 5, 7, and 2.
2. 2All 3 of Prism A's edge lengths stay together to make one product for $V_A$ .
Common mistake. A common mistake is entering only 2 of the 3 edge lengths, such as writing $V_A = 5 \times 7$ and leaving out the third dimension 2. All 3 edge lengths must appear as separate factors.
The idea underneath. For volume of a right rectangular prism, all 3 edge lengths are multiplied together as one product. The order of the factors does not matter, but none of the 3 edge lengths can be left out.
2On level
Look at the right rectangular prism below. Each small cube has edge length $\frac{1}{2}$ unit.
Front View
Back View
Emma and Carlos wrote the expressions shown below.
$\displaystyle \begin{array}{l} \text{Emma: } 2 \times 1 \times \frac{3}{2} \\[0.5em] \text{Carlos: } 3 \times 1 \times 2 \end{array}$
Select who wrote a correct length × width × height expression for the prism.
A
Emma
B
Carlos
C
Both students
D
Neither student
Show the answer and how to solve it
Emma is correct because the prism's edge lengths are $2$ , $\frac{3}{2}$ , and $1$ unit, and Emma's expression uses exactly those three lengths.
1. 1Look at the prism and count the small cubes along its three edges. The edge counts are 4, 3, and 2 cubes.
2. 2Each small cube edge is $\frac{1}{2}$ unit, so convert each cube count to an actual edge length.
3. 3The prism's three edge lengths are $2$ , $\frac{3}{2}$ , and $1$ unit. A correct length × width × height expression can list those three factors in any order.
4. 4Compare each student's factors to the true edge lengths. Emma uses $2$ , $1$ , and $\frac{3}{2}$ — a reordering of the true edge lengths, so Emma is correct. Carlos uses $3$ , $1$ , and $2$ ; the factor $3$ is the cube count for the depth edge, not the edge length $\frac{3}{2}$ .
5. 5Only Emma wrote a correct length × width × height expression.
Common mistake. A common mistake is to write a cube count instead of an edge length. The depth edge has $3$ cubes; since each cube edge is $\frac{1}{2}$ unit, that edge is $\frac{3}{2}$ units long, not $3$ .
The idea underneath. In a packing model, each factor in length × width × height must be an actual edge length, so cube counts must be converted using the size of one small cube.
3On level
Some of these objects are one rectangular prism, and some are made from two rectangular prism parts.
Select every object whose volume you find by adding the volumes of two prism parts.
A
A storage bench has a 5 ft by 2 ft by 2 ft base and a 3 ft by 2 ft by 1 ft top section.
B
A packing crate measures 4 ft by 3 ft by 2 ft.
C
A bookcase has a 4 ft by 1 ft by 4 ft lower section with a 2 ft by 1 ft by 2 ft upper section attached on top.
D
A toy chest is 3 ft long, 2 ft wide, and 2 ft tall.
Show the answer and how to solve it
Select the storage bench and the bookcase, because each one is made from 2 prism parts whose volumes must be added. Leave the packing crate and the toy chest unchecked, because each is a single rectangular prism.
1. 1For each object, decide whether it describes 1 box-shaped solid or 2 attached box-shaped parts. Select an object only when it is made from 2 prism parts.
2. 2The packing crate, 4 ft by 3 ft by 2 ft, and the toy chest, 3 ft long by 2 ft wide by 2 ft tall, each describe 1 rectangular prism. You find each volume with one rectangular prism volume formula, so leave those two unchecked.
3. 3The storage bench has a base and a top section, and the bookcase has a lower section with an upper section attached on top. Each of those solids is made of 2 rectangular prism parts, so you find each part's volume and add. Select the storage bench and the bookcase.
Common mistake. A common mistake is checking the packing crate because its three dimensions seem complex, or leaving the storage bench unchecked because both parts share the same 2 ft width. Count prism parts, not shared measurements: 2 attached sections still need part volumes added, so the storage bench should be selected.
The idea underneath. The strategy depends on the structure of the solid, not on how many numbers are listed: 1 rectangular prism uses 1 volume, but 2 prism parts must be combined. Check an object only when it is built from 2 prism parts.

4Stretch

A storage crate is a right rectangular prism that is

1\frac{1}{3}

ft long,

\frac{3}{4}

ft wide, and

\frac{5}{8}

ft high. Marcus's work for the volume is shown below.

Step 1	$\displaystyle 1\,\frac{1}{3} = \frac{4}{3}$
Step 2	$\displaystyle \frac{4}{3} \times \frac{3}{4} \times \frac{5}{8}$
Step 3	$\displaystyle = \frac{4 \times 3 \times 5}{3 \times 4 \times 8}$
Step 4	$\displaystyle = \frac{60}{96}$
Step 5	$\displaystyle = \frac{5}{6} \text{ cubic feet}$

Select the statement that best explains Marcus's mistake.

The final fraction was simplified incorrectly.

The dimensions were added instead of multiplied.

A mixed number was converted incorrectly.

The work finds the area of one face, not the volume.

Show the answer and how to solve it

The correct answer is 'The final fraction was simplified incorrectly' because Marcus correctly reached $\frac{60}{96}$ but then reduced it to $\frac{5}{6}$ instead of $\frac{5}{8}$ .

1Read through Marcus's work from top to bottom to locate the first step that contains an error.
2Step 1 correctly converts $1\frac{1}{3}$ to $\frac{4}{3}$ . Steps 2 and 3 correctly write the product of all three dimensions and evaluate: $4 \times 3 \times 5 = 60$ and $3 \times 4 \times 8 = 96$ , so $\frac{60}{96}$ in Step 4 is accurate. The error first appears in Step 5.
3Check the simplification in Step 5. The greatest common factor of 60 and 96 is 12, so the fraction must reduce to $\frac{5}{8}$ , not $\frac{5}{6}$ .
4Because the mixed-number conversion and all earlier fraction steps are correct, the only error is in the final simplification of $\frac{60}{96}$ . The best explanation is that the final fraction was simplified incorrectly.

Common mistake. A common mistake is applying different divisors to the numerator and denominator when simplifying. Marcus divided 60 by 12 to get 5 correctly, but divided 96 by 16 instead of 12, producing $\frac{5}{6}$ rather than the correct $\frac{5}{8}$ .

The idea underneath. In error analysis, find the first incorrect step, because later arithmetic can be correct from a wrong starting value.