Volume with Fractional Edge Lengths

Grade 6 Geometry — Standard 6.G.A.2

In this lesson:

• Explain why packing with unit fraction cubes gives V = lwh
• Apply V = lwh and V = Bh with fractional edges
• Solve real-world volume problems with fractions

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Explain why packing with unit fraction cubes gives
2. Apply and with fractional edges
3. Solve real-world volume problems with fractions

Does V = lwh Work for Fractions Too?

For whole-number edges you already know:

Example: , ,

Does the same formula work when , , or is a fraction?

Choosing the Right Unit Fraction Cube

To pack a prism with fractional edges, we need a cube that fits all three dimensions evenly.

Prism:

Find the packing cube edge: LCM of denominators 1, 2, 3 → LCM = 6

Packing cube edge =

Counting Cubes Along Each Edge

Divide each edge by the packing cube edge :

• Length: cubes
• Width: cubes
• Height: cubes

Packing the Prism — Finding the Volume

Total cubes: 72. Cube edge =

Step 3: Volume of each small cube

Step 4: Total volume

Formula check: ✓

Check: Try the Packing Argument Yourself

Prism: — packing cube edge =

• Count along each edge: 12, 2, 1 → total = ?
• Each cube volume:

Find total volume, then verify with the formula.

Packing Confirms the Formula Every Time

cubes; cubic unit

Formula check: ✓

counts unit fraction cubes automatically — no new formula needed for fractions.

Two Formulas for Rectangular Prism Volume

• — multiply all three edge lengths
• — find base area first: , then

, , are perpendicular edge lengths — not diagonals.

Example 1: Pure Fraction Dimensions

Find the volume: , ,

Example 2: Mixed Number Dimensions

Find the volume: , ,

Step 1: Convert mixed number to improper fraction

Step 2: Apply the formula

Your Turn: Apply the Formula

Find the volume of this prism:

, ,

Hint: No mixed numbers here — set up and multiply.

Try it on your own, then advance for the answer.

Guided Practice Answer and Shortcut

Or simplify first: , then

Real-World Example: Fish Tank Volume

A fish tank measures ft × ft × ft.

Find the volume.

Solve for Height Using V = Bh

Gift box: base ft × ft; volume = ft³

Your Turn: Find Each Volume

Include units in your final answer.

1. cm, cm, cm

2. ft, ft, ft

Try both before advancing to the answers.

Checking Your Practice Problem Answers

Problem 1:

Problem 2:

What to Remember from This Lesson

✓ works for fractional edges — packing with unit fraction cubes proves it

✓ Always convert mixed numbers to improper fractions before multiplying

✓ is useful when the base area is given directly

Watch out: Multiply the edges — do not add them.

Watch out: . Convert first:

Next Lesson: Surface Area with Fractions

Coming up — Standard 6.G.A.4

• Same fractional dimensions applied to surface area
• Total area of all six faces of a rectangular prism
• Fraction multiplication skills from today carry forward