In Grade 3 you measured area with unit squares: how many squares cover a flat shape. Volume is the same idea one step up: how many cubes fill a solid.
Idea clave
A unit cube is 1 unit long, 1 unit wide and 1 unit high; its volume is 1 cubic unit. The volume of a solid is the number of unit cubes that fill it, with no gaps and no overlaps.
Por qué funciona
Volume measures the space inside. Cubes stack and pack with no gaps, so counting them measures space the way counting squares measures flat area. Packing a box layer by layer is a quick way to count: every layer holds the same number of cubes.
Worked examples
Ejemplo resuelto: Count by layers
A box is filled with unit cubes: 4 cubes long, 3 cubes wide and 2 cubes high. What is its volume?
- One layer: 4 rows of 3, which is $4 \times 3 = 12$ cubes.
- There are 2 layers: $12 + 12 = 24$ cubes.
Respuesta: 24 cubic units
Ejemplo resuelto: Cubes you cannot see
A cube-shaped box is 3 cubes on every edge. How many unit cubes fill it?
- One layer: $3 \times 3 = 9$ cubes.
- 3 layers: $9 \times 3 = 27$ cubes. From one corner you can see only 19 of them; the other 8 are hidden behind and inside.
Respuesta: 27 cubic units
Error común
Counting only the cubes you can see on the outside of a picture. A $3 \times 3 \times 3$ cube shows 19 cubes from one corner, but holds 27. Check: count by layers, cubes in one layer times the number of layers.