12.4 Volume by Unit Cubes & Additive Volume of Composite Solids

Volume by Unit Cubes & Additive Volume of Composite Solids

Section 12.1 supplied the volume formulas for prisms, cylinders, pyramids, cones, and spheres. This section covers the Level M standards underneath them: understanding volume as a count of unit cubes, deriving $V = lwh$ from that count, and finding the volume of solids composed of two rectangular prisms by adding.

Volume Is a Count of Unit Cubes

Volume is the number of unit cubes — cubes 1 unit on every edge — needed to fill a solid with no gaps and no overlaps.

A cube 1 centimeter on each edge has a volume of 1 cubic centimeter (1 cm³). Fill a box with 60 of them and its volume is 60 cubic centimeters.

Filling Layer by Layer

A 4 × 3 × 2 rectangular prism:

Layer 1 (bottom):        Layer 2 (top):
┌───┬───┬───┬───┐        ┌───┬───┬───┬───┐
├───┼───┼───┼───┤        ├───┼───┼───┼───┤
├───┼───┼───┼───┤        ├───┼───┼───┼───┤
└───┴───┴───┴───┘        └───┴───┴───┴───┘
  4 × 3 = 12 cubes         4 × 3 = 12 cubes

Total: 2 layers × 12 cubes = 24 cubic units

Two readings of the same count give the two formulas:

$V = Bh$ is the more powerful form because it generalizes: any prism's volume is its base area times its height, whether the base is a triangle, a trapezoid, or a circle. A cylinder is just a prism whose base is a circle, which is why $V = \pi r^2 h$ — the $\pi r^2$ is $B$.

Why the unit is cubed. $\text{cm} \times \text{cm} \times \text{cm} = \text{cm}^3$. An answer whose unit is not cubed is not a volume. This is the fastest way to catch the classic error of reporting surface area when the question asked for volume.

Counting Cubes on TABE

Items show a stack of cubes and ask for the volume. Two habits keep you accurate:

1. Count by layers, not by visible faces. Cubes hidden behind others still count.
2. Check for missing cubes. If the drawing shows a notch, subtract the cubes that are absent.

A drawing shows a solid built as a 5 × 4 base, 3 layers tall, with a 2 × 2 × 1 notch removed from one top corner.

Full block: $5 \times 4 \times 3 = 60$ cubes. Notch: $2 \times 2 \times 1 = 4$ cubes. Volume: $60 - 4 = \mathbf{56}$ cubic units.

Additive Volume: Composite Solids

The Level M standard says it directly: find volumes of solids composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts.

The procedure:

1. Slice the solid into rectangular prisms — one horizontal or vertical cut usually suffices.
2. Find each prism's dimensions, deducing any that are not labeled.
3. Compute each volume.
4. Add.

Worked example. A loading dock is poured as an L-shaped concrete pad. The lower section is 20 ft × 12 ft × 1 ft. A raised section sits on part of it, 8 ft × 12 ft × 2 ft. What is the total concrete volume?

• Lower section: $20 \times 12 \times 1 = 240$ cu ft
• Raised section: $8 \times 12 \times 2 = 192$ cu ft
• Total: 432 cubic feet

Converting to the unit concrete is ordered in: $432 \div 27 = \mathbf{16}$ cubic yards.

Worked example with a deduced dimension. A storage unit is a 10 ft × 8 ft × 9 ft room with a 3 ft × 8 ft × 4 ft alcove added along one wall.

• Main room: $10 \times 8 \times 9 = 720$ cu ft
• Alcove: $3 \times 8 \times 4 = 96$ cu ft
• Total: 816 cubic feet

The overlap warning. "Non-overlapping" is in the standard for a reason. If you slice a solid so that a region belongs to both pieces, you will count it twice. Draw the cut line and confirm that every cubic inch belongs to exactly one piece.

Subtracting: Hollow Solids

Some items need the reverse operation.

A concrete pipe is 10 ft long with an outer diameter of 4 ft and an inner diameter of 3 ft. How much concrete? (Use $\pi \approx 3.14$.)

• Outer cylinder: $3.14 \times 2^2 \times 10 = 125.6$ cu ft
• Inner void: $3.14 \times 1.5^2 \times 10 = 70.65$ cu ft
• Concrete: $125.6 - 70.65 = 54.95$ cubic feet

Volume and Scaling

If every linear dimension is multiplied by $k$:

Doubling every dimension of a box multiplies its volume by 8, not by 2. A 2 × 3 × 4 box holds 24 cubic units; a 4 × 6 × 8 box holds 192, and $192 \div 24 = 8 = 2^3$.

This is why a container twice as tall, wide, and deep holds eight times as much — a result that regularly appears as a TABE reasoning item and that almost everyone underestimates on instinct.

Volume and Capacity

Volume in cubic units converts to liquid capacity:

A tank measures 3 ft × 2 ft × 2 ft. Its volume is 12 cubic feet, which holds about $12 \times 7.48 \approx \mathbf{90}$ gallons.