Volume

5.I.1Counting unit cubes01/04

Volume by counting unit cubes.

Step01of 04

Volume measures the amount of space inside a 3D shape, in CUBIC units. The simplest way to start: count how many UNIT CUBES (1 × 1 × 1) fit inside.

5.I.1 — Volume by counting unit cubes.

Step 01 of 04

Volume measures the amount of space inside a 3D shape, in CUBIC units. The simplest way to start: count how many UNIT CUBES (1 × 1 × 1) fit inside.

Step 02 of 04

Worked example. A rectangular prism made of unit cubes is 3 wide, 2 deep, 4 tall. Total cubes?

Count the cubes layer by layer. Each layer is 3 × 2 = 6 cubes. There are 4 layers.

6 \times 4 = 24 cubic units

Step 03 of 04

This counting method makes the volume formula visible: $V = l \times w \times h$ . The factors are exactly the dimensions.

Step 04 of 04

Why "cubic units"? Each cube is 1 × 1 × 1 unit, so its volume is 1 cubic unit. Counting cubes literally counts cubic units.

Key insight

Volume = number of unit cubes that fit. Layer by layer is the easy way to count: rectangular base × number of layers. This becomes the formula V = l × w × h.

5.I.2 — Volume of rectangular prism — formula.

Step 01 of 04

Volume formula.

V = l \times w \times h

Length × width × height. Multiply all three dimensions to get the volume in cubic units.

Step 02 of 04

Worked example. A box is 5 cm long, 3 cm wide, 4 cm tall.

V = 5 \times 3 \times 4 = 60 cubic cm

Step 03 of 04

Equivalent formula. $V = B \times h$ , where $B$ is the area of the base. Same answer, just compute the base area first.

B = 5 \times 3 = 15, V = 15 \times 4 = 60

Step 04 of 04

Find a missing dimension. Volume 120 cu cm, length 6, width 4. Find height.

120 = 6 \times 4 \times h \Rightarrow 24 h = 120 \Rightarrow h = 5 cm

Key insight

V = l × w × h. Or V = base area × height. Volume in CUBIC units. To find a missing dimension, divide volume by the product of the known dimensions.

5.I.3 — Volume of composite figures.

Step 01 of 04

A composite figure is built from two or more rectangular prisms joined together. Find the volume of each piece separately, then add them.

Step 02 of 04

Worked example — L-shape. An L-shaped block has two rectangular pieces. Piece A: 4 × 3 × 5 = 60. Piece B: 2 × 3 × 5 = 30.

V_{total} = 60 + 30 = 90 cu units

Step 03 of 04

Subtract instead of add. Sometimes it's easier to compute the volume of a "complete" prism, then subtract the missing piece.

A 6 × 4 × 5 prism with a 2 × 2 × 2 cube removed from the corner: $V = 6 \cdot 4 \cdot 5 - 2^{3} = 120 - 8 = 112$ .

Step 04 of 04

Identify the pieces clearly. Sketch the shape, label dimensions, decide on a split. Different splits give the same final answer (volumes don't depend on which way you carve up the figure).

Key insight

Composite figure → split into rectangular prisms, compute each volume, ADD. Or compute the bounding prism and SUBTRACT the missing part. Either approach works.

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5.I.1 – 5.I.3 · CCSS 5.MD.C · Full curriculum →