Grades 3–5 · Mathematics
How do unit cubes measure a rectangular solid's volume?
A box takes up space in three directions: length, depth and height. Volume measures how much of that three-dimensional space it occupies. A unit cube has edges one unit long and volume one cubic unit. Equal cubes can measure a solid when they fill it with no gaps or overlaps.
In the diagramVolume counts unit cubes in solid space3 long2 deep2 highEach small cube is 1 cubic unit.3 × 2 × 2 = 12 cubic unitsCubes fill every layer without gaps or overlaps.
The solid here is three cubes long, two cubes deep and two cubes high. Its bottom layer has two rows of three cubes, so that layer contains six cubes. Another layer of six sits above it. There are twelve cubes altogether, including cubes hidden behind the front ones.
The layer count gives 3 × 2 × 2 = 12 cubic units. Each factor measures one perpendicular direction. Counting only visible faces would miss hidden cubes and mix up a surface with the space inside. Square units measure flat covering; cubic units measure filled solid space.
About this explanation
Starting knowledge: Multiply small whole numbers and distinguish a flat array from a three-dimensional solid.
Sources
- Common Core State Standards Initiative — Grade 5: Unit cubes and rectangular-prism volume