Packing Cubes With Fractional Edge Lengths Inside A Box

📝 Practice Worksheets

Cubes with Fractional Edges Fractional Cubic Units Rectangular Prism Story Problems Non-Whole-Number Dimensions

🛠️ Related Tools

Rectangular Prism Volume Explorer Volume Builder

🔑 Key Concepts

The box measures \(3\) feet by \(2\frac{1}{2}\) feet by \(2\) feet.
Each small cube has an edge length of \(\frac{1}{2}\) foot.
Six cubes fit along the \(3\)-foot length because \(3\div\frac{1}{2}=6\).
Five cubes fit along the \(2\frac{1}{2}\)-foot width because \(2\frac{1}{2}\div\frac{1}{2}=5\).
Four cubes fit along the \(2\)-foot height because \(2\div\frac{1}{2}=4\).
Each layer contains \(6\times5=30\) cubes.
Four layers contain \(30\times4=120\) cubes.

✏️ Interactive Worked Example

Pack \(\frac{1}{2}\)-Foot Cubes into a \(3\times2\frac{1}{2}\times2\)-Foot Box

Filled: 0 of 4 layers

Number of layers

Each layer is \(\frac{1}{2}\) foot high and contains \(30\) congruent cubes.

Cubes per layer

6 × 5

30 cubes

Filled height

h = 0 ft.

0 layers

Cubes packed

30 × layers

0 cubes

Step 1: Count Cubes Along Each Dimension

Divide each box dimension by the \(\frac{1}{2}\)-foot edge length of one cube.

\(\displaystyle 3\div\frac{1}{2}=6\)

\(\displaystyle 2\frac{1}{2}\div\frac{1}{2}=5\)

\(\displaystyle 2\div\frac{1}{2}=4\)

The arrangement is \(6\) cubes by \(5\) cubes by \(4\) cubes.

Step 2: Find the Number in One Layer

One horizontal layer is \(6\) cubes long and \(5\) cubes wide.

\(6\times5=30\)

\(\boxed{30\text{ cubes per layer}}\)

Each layer is \(\frac{1}{2}\) foot high.

Step 3: Stack Four Layers

Four half-foot layers fill the \(2\)-foot height.

\(30\times4=120\)

\(\boxed{120\text{ cubes}}\)

Therefore, \(120\) half-foot cubes can be packed inside the box.

A Second Method: Divide the Two Volumes

Find the volume of the large rectangular prism:

\(V=lwh\)

\(\displaystyle V=3\left(2\frac{1}{2}\right)(2)\)

\(\displaystyle V=3\left(\frac{5}{2}\right)(2)\)

\(\boxed{V=15\text{ ft}^3}\)

Find the volume of one half-foot cube:

\(\displaystyle V=\left(\frac{1}{2}\right)^3\)

\(\displaystyle V=\frac{1}{8}\text{ ft}^3\)

Divide the box volume by the cube volume:

\(\displaystyle 15\div\frac{1}{8}=15\times8\)

\(\boxed{120\text{ cubes}}\)

Both methods agree. Multiplying the number of cubes along the three dimensions gives \(6\times5\times4=120\), while dividing the total volume by \(\frac{1}{8}\) cubic foot also gives \(120\).

Why Does Dividing by \(\frac{1}{8}\) Mean Multiplying by \(8\)?

A cube with a \(\frac{1}{2}\)-foot edge has a volume of:

\(\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}\text{ ft}^3 \)

This means that eight half-foot cubes fit inside one cubic foot:

\(\displaystyle 1\div\frac{1}{8}=8 \)

Since the box contains \(15\) cubic feet, it can hold \(15\times8=120\) cubes of this size.

🧠 Math Vocabulary

Rectangular prism: A three-dimensional solid with six rectangular faces.
Packing: Filling a container with smaller objects without gaps or overlaps.
Cube: A rectangular prism with congruent square faces and equal edge lengths.
Edge length: The length of one edge of a three-dimensional solid.
Fractional edge length: An edge length represented by a fraction, such as \(\frac{1}{2}\) foot.
Dimension: A measurable direction, such as length, width, or height.
Volume: The amount of three-dimensional space contained inside a solid.
Fractional cubic unit: A portion of a full cubic unit, such as \(\frac{1}{8}\text{ ft}^3\).
Layer: One complete level of congruent cubes covering the base.
Congruent cubes: Cubes with the same shape and dimensions.
Reciprocal: A number that produces \(1\) when multiplied by the original number.
Divide by a fraction: Multiply by the reciprocal of the fraction.
Capacity: The amount that a three-dimensional container can hold.

💡 Main Idea

To determine how many fractional-edge cubes fit inside a rectangular prism, divide each box dimension by the cube’s edge length and multiply the resulting counts. For this box, \(6\) cubes fit along the length, \(5\) fit along the width, and \(4\) fit along the height, giving \(6\times5\times4=120\) cubes.

📚 What You Should Already Know

Students should know how to find the volume of cubes and rectangular prisms, divide whole and mixed numbers by fractions, multiply by reciprocals, and interpret volume using cubic units.

🚀 What Comes Next

Students can extend this reasoning to cubes with other fractional edge lengths, containers with decimal dimensions, incomplete packing arrangements, and problems requiring them to determine the greatest number of cubes that can fit without cutting or overlapping.

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