Same Volume, Different Shapes
Same Volume, Different Shapes is a Grade 5 math skill from Illustrative Mathematics Chapter 1 (Finding Volume) that establishes the concept that two solid objects composed of the same number of unit cubes have equal volumes regardless of their shape. A tall thin tower and a short wide block can both have a volume of 8 cubic units if each is built from 8 unit cubes. Volume depends on the total cube count, not appearance.
Key Concepts
If two solid objects are composed of the same number of identical unit cubes, their volumes are equal, regardless of their shape. If object A has $N$ cubes and object B has $N$ cubes, then $Volume A = Volume B$.
Common Questions
Can two shapes with different appearances have the same volume?
Yes. If two solid objects are built from the same number of identical unit cubes, their volumes are equal even if they look completely different. For example, a 2×2×2 block and an 8-cube tall tower both have a volume of 8 cubic units.
Why does volume depend on unit cubes and not shape?
Volume measures the total amount of space a 3D object occupies, which is counted by how many unit cubes fill it. The shape or arrangement of those cubes does not change their count, so volume stays the same regardless of how the cubes are arranged.
What chapter covers same volume different shapes in Illustrative Mathematics Grade 5?
Same Volume, Different Shapes is covered in Chapter 1 of Illustrative Mathematics Grade 5, titled Finding Volume.
How can students demonstrate same volume with different shapes?
Students can rearrange the same set of unit cubes into different configurations—a tall column, an L-shape, or a flat layer—and observe that counting the cubes always gives the same total, demonstrating that volume is preserved.
What is an example of two different shapes with the same volume?
An L-shaped figure made of 6 cubes and a flat 2×3×1 rectangle also made of 6 cubes both have a volume of 6 cubic units. Their shapes look different, but they contain the same number of unit cubes.