Volume Of A Rectangular Prism - Cubes With Fracitonal Edge Lengths

🔑 Key Concepts

• The structure is composed of \(12\) congruent cubes.
• Each small cube has an edge length of \(\frac{1}{2}\) inch.
• The volume of one small cube is \(\left(\frac{1}{2}\right)^3=\frac{1}{8}\) cubic inch.
• The entire structure has a volume of \(12\times\frac{1}{8}=\frac{3}{2}\) cubic inches.
• The structure can also be treated as one rectangular prism with dimensions \(1\frac{1}{2}\) inches, \(\frac{1}{2}\) inch, and \(2\) inches.
• Both methods produce the same volume: \(1\frac{1}{2}\) cubic inches.

✏️ Worked Example

The Structure

The prism contains \(12\) small cubes arranged in:

\(3\) cubes across
\(1\) cube deep
\(4\) cubes high

Each small cube has an edge length of \(\frac{1}{2}\) inch.

Method 1: Find the Prism Dimensions

Multiply the number of cubes along each direction by the edge length of one cube.

\(\displaystyle l=3\left(\frac{1}{2}\right) =\frac{3}{2} =1\frac{1}{2}\text{ in.} \)

\(\displaystyle w=1\left(\frac{1}{2}\right) =\frac{1}{2}\text{ in.} \)

\(\displaystyle h=4\left(\frac{1}{2}\right) =\frac{4}{2} =2\text{ in.} \)

Substitute the dimensions into \(V=lwh\):

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

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

\(\displaystyle V=\frac{3}{2} \)

Volume:
\(\boxed{1\frac{1}{2}\text{ in}^3}\)

Method 2: Use the Twelve Small Cubes

First, find the volume of one small cube.

\(V=s^3\)

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

\(\displaystyle V= \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \)

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

There are \(12\) identical cubes:

\(\displaystyle 12\left(\frac{1}{8}\right) =\frac{12}{8} \)

\(\displaystyle \frac{12}{8} = \frac{3}{2} = 1\frac{1}{2} \)

Total volume:
\(\boxed{1\frac{1}{2}\text{ in}^3}\)

Why Both Methods Produce the Same Answer

Using the dimensions of the complete rectangular prism:

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

\(\displaystyle V=\frac{3}{2}\text{ in}^3 \)

Using the volume of each small cube:

\(\displaystyle V= 12\left(\frac{1}{8}\right) \)

\(\displaystyle V=\frac{3}{2}\text{ in}^3 \)

The first method treats the structure as one rectangular prism. The second method adds the volumes of its \(12\) congruent cubes. Both methods measure the same three-dimensional space, so their answers must agree.

Why One Small Cube Is Only \(\frac{1}{8}\) Cubic Inch

An edge length of \(\frac{1}{2}\) inch does not create a cube with a volume of \(\frac{1}{2}\) cubic inch. All three dimensions are one-half inch:

\(\displaystyle \frac{1}{2}\text{ in.} \times \frac{1}{2}\text{ in.} \times \frac{1}{2}\text{ in.} \)

\(\displaystyle \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \)

\(\boxed{\frac{1}{8}\text{ in}^3}\)

Eight cubes with edge lengths of \(\frac{1}{2}\) inch would fit inside one cube measuring \(1\) inch on every edge.

🧠 Math Vocabulary

• Rectangular prism: A three-dimensional solid with six rectangular faces.
• Cube: A rectangular prism with six congruent square faces and equal edge lengths.
• Edge length: The length of one edge of a three-dimensional solid.
• Dimension: A measurable direction, such as length, width, or height.
• Fractional dimension: A length, width, or height represented by a fraction.
• Congruent cubes: Cubes with exactly the same shape and dimensions.
• Volume: The amount of three-dimensional space contained inside a solid.
• Cubic unit: A unit used to measure volume.
• Fractional cubic unit: A portion of a full cubic unit, such as \(\frac{1}{8}\text{ in}^3\).
• Unit cube: A cube with edge lengths of one unit and a volume of one cubic unit.
• Composite structure: A three-dimensional figure made from two or more smaller solids.
• Improper fraction: A fraction whose numerator is greater than or equal to its denominator.
• Mixed number: A number containing a whole-number part and a fractional part.
• Product: The result of multiplication.

💡 Main Idea

A structure made from fractional cubic units can be analyzed in two ways. Find the dimensions of the complete rectangular prism and use \(V=lwh\), or find the volume of one small cube and multiply by the number of cubes. In this example, both methods show that the volume is \(1\frac{1}{2}\) cubic inches.

📚 What You Should Already Know

Students should know how to find the volume of cubes and rectangular prisms, multiply fractions, simplify improper fractions, rename improper fractions as mixed numbers, and label volume using cubic units.

🚀 What Comes Next

Students can extend this reasoning to structures with different fractional edge lengths, incomplete arrangements of cubes, missing cubes, composite rectangular prisms, and multistep volume story problems.

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