Volume with unit cubes

What the tool shows

Volume is the amount of space a solid takes up. It is measured in cubic units: one unit cube is 1 unit long, 1 unit wide and 1 unit high. On this page, the steppers set the length (1 to 6), width (1 to 5) and height (1 to 5) of a rectangular prism drawn from unit cubes. Each layer has its own color. Show the layers pulls them apart so a child can see that every layer is the same.

The readout gives V = l × w × h with the numbers filled in, and explains it as layers: each layer is length × width cubes, and there are height layers. Hide the total turns this off so the child counts first.

Why build with cubes

The formula V = l × w × h is easy to memorize and easy to misuse. Counting cubes shows where it comes from. The bottom layer is an array of length × width cubes, which is the area of the base, and stacking the layers multiplies that by the height. Common Core asks grade 5 students to measure volume by counting unit cubes (5.MD.C.4) and then connect it to multiplication (5.MD.C.5). Seeing that a 2 by 3 by 4 box and a 1 by 4 by 6 box both hold 24 cubes also gives practice with factors.

Common mix-ups

• Counting only the cubes you can see. Hidden cubes in the middle and at the back still count, which is why counting by layers works better.
• Adding the dimensions instead of multiplying them. A 2 by 3 by 4 box holds 24 cubes, not 9.
• Thinking a turned box has a new volume. Turning a box changes which side is the length, but not how many cubes it holds.
• Using square units. Volume is in cubic units; area is in square units.

A 10-minute volume lesson

• Turn on Hide the total and Show the layers. Build a 3 by 2 base with a height of 1 and count the cubes.
• Add a layer at a time. After each one, ask for the new total before pressing Show me.
• Ask the child to describe a quick way to count, which leads to layer × number of layers.
• Press Challenge and find two different boxes with the target volume.
• End by writing the volume of the last box as l × w × h.

Where it fits by grade

• Grade 5: understand volume and unit cubes (5.MD.C.3), count unit cubes (5.MD.C.4), and use V = l × w × h and V = B × h (5.MD.C.5).
• Grade 6: volume of prisms with fractional edge lengths (6.G.A.2).
• England: Year 5 estimates volume by building cuboids from 1 cm³ blocks, and Year 6 calculates and compares the volume of cubes and cuboids using standard units.

Learning more

Times tables and area are both on Eurekami's skill map, and Daily Practice brings them back when they start to fade. Lessons draw diagrams live from the numbers in each question, and when a word problem about boxes is tricky, the AI tutor asks guiding questions so the student does the thinking.