What is a perfect cube?
A perfect cube is what you get when you multiply a whole number by
itself three times. 5 × 5 × 5 = 125, so 125 is a perfect cube.
The shorthand is 5³ — read as five cubed.
The same picture has a second reading. A cube with side 5 is built
from 5 × 5 × 5 = 125 unit cubes. The number 125 is the volume
of that cube. So a perfect cube is a count and a shape at the same
time: the count of unit cubes, and the cube they fill.
The cube root runs the picture in reverse. ³√125 = 5 because 5
is the side length of a cube with volume 125. The little 3 on the
radical reads as the side length of a cube with volume….
Same cube, two views. The stack shows the 27 unit cubes packed
inside. The unfolded net shows the six square faces — six because a
cube has six sides, square because every face of a cube is a square.
See it: build a cube
Try it
Drag the slider. Watch the cube grow.
Use the slider, or focus it and use ←/→ to change the side length.
The toggle reads the same stack two ways: n × n × n = n³
(multiplication) or ³√(n³) = n (cube root). Same picture, two
equations.
The recall band stops at 5³ = 125 because cubes grow fast. 6³ is
already 216, and 10³ is 1 000. Five values are worth memorizing;
beyond that, work it out.
Where it shows up in real life
Stack hay bales in a square grid, then stack the same grid up to the
same height: a 3 × 3 × 3 stack has 27 bales. A 4 × 4 × 4 stack
has 64. Anything packed into a cube — bales in a barn, sugar cubes
in a box, server racks in a 3 × 3 × 3 cluster — the count is a
perfect cube.
The same logic runs in reverse: if you have 64 boxes and you want
to stack them into a perfect cube, the side is ³√64 = 4. With 50
boxes you can't make a perfect cube — 50 sits between 27 = 3³ and
64 = 4³, so ³√50 is between 3 and 4.
Worksheet
These aren't graded. Get them right, get them wrong — the goal is to
build recall for 1³ through 5³, and to keep cubes and squares
straight in your head.
Question 1 of 3
Try it
What is 4³?
Multiple choice: what is four cubed? Four answer cards: twelve, sixty-four, sixteen, eighty-one.Perfect cubes
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
Going further
Cube roots of non-perfect cubes — like ³√50 — sit between two
whole numbers. ³√50 is between 3 and 4, closer to 4.
Estimating those values is the next step.
Cubes are one shape; rectangular boxes are another. A box with length
l, width w, and height h has volume l × w × h. When all three
are equal, the box is a cube and the volume is s³. Same idea, three
dimensions, different shapes.