Key takeaway: Volume is the amount of three-dimensional space a solid occupies, measured in cubic units, and for a box it equals the product of its three edge lengths.
Grade 66.G.A.2Geometry and Measurement
Definition of Volume
The volume of a solid tells how much space it fills. To measure it, imagine packing the solid with unit cubes, each one unit long, wide and tall, and counting them. That count, in cubic units, is the answer.
For a box, you can stack identical layers: one layer holds length times width cubes, and the height says how many layers there are. Multiplying all three gives the total. Cubic units, such as cubic inches, appear because three lengths are multiplied. Capacity, like how much water a tank holds, is the same idea applied to the inside space.
Formula and notation
For a box with length \(\ell\), width \(w\) and height \(h\):
\[V = \ell \times w \times h \qquad\text{or}\qquad V = (\text{base layer}) \times h\]
Here the base layer is the number of squares on the floor of the box. Units are cubed: \(\text{in}^3\), \(\text{ft}^3\), \(\text{cm}^3\).
Key facts
- Counting cubes measures this quantity, counting squares measures surface, and counting steps measures length.
- Doubling all three edges of a box makes the volume eight times larger.
- Fractional edge lengths work with the same multiplication rule.
- Cubic units never appear as plain units of length.
Volume example
A moving box measures 30 inches long, 18 inches wide and 16 inches tall. Find its volume, and then how many 2-inch cubes of packing foam would fill it.
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One layer holds \(30 \times 18 = 540\) cubes of side 1 inch.
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Sixteen layers: \(540 \times 16 = 8{,}640\) cubic inches.
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A 2-inch cube has volume \(2 \times 2 \times 2 = 8\), so \(8{,}640 : 8 = 1{,}080\) cubes.
The volume is 8,640 cubic inches, which holds 1,080 foam cubes.
Common mistake
Watch out
Tia finds the volume by adding the edges: \(30 + 18 + 16 = 64\) cubic inches.
Why it is wrong
Adding the edges gives a length, not a count of cubes. The right move multiplies the three dimensions: \(30 \times 18 \times 16 = 8{,}640\). A box this large clearly holds far more than 64 one-inch cubes.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Geometry and Measurement