What is a perfect square?
A perfect square is what you get when you multiply a whole number
by itself. 7 × 7 = 49, so 49 is a perfect square. The shorthand
is 7² — read as seven squared.
The same picture has a second reading. A square with side 7 is made
of 7 × 7 = 49 unit tiles. The number 49 is the area of that
square. So a perfect square is a count and a shape at the same time:
the count of tiles, and the square they fill.
The square root runs the picture in reverse. √49 = 7 because
7 is the side length of a square with area 49. The radical sign
√ reads as the side length of a square with area….
Try it
Drag the slider. Watch the square grow.
Use the slider, or focus it and use ←/→ to change the side length.
The toggle reads the same square two ways: n × n = n²
(multiplication) or √(n²) = n (square root). Same picture, two
equations.
Which numbers are perfect squares?
Most numbers aren't perfect squares. Between 49 = 7² and 64 = 8²
sit 50, 51, 52, …, 63 — fourteen integers, none of them perfect.
The perfect squares thin out as you go: the gap between consecutive
ones grows by 2 every step.
Try it
Tap any number. The panel tells you whether it's a perfect square.
Perfect square
49 = 7 × 7·Square root: √49 = 7
Perfect squares on this strip1 · 4 · 9 · 16 · 25 · 36 · 49 · 64 · 81 · 100 · 121 · 144 · 169 · 196
Glowing dots are perfect squares. Tap a dim one to see what sits between them.
A non-perfect square sits between two perfect ones — 50 is between
49 = 7² and 64 = 8², so √50 is between 7 and 8. Estimating
those in-between roots comes later.
Where it shows up in real life
A square patio in Red Deer needs the right number of tiles. Side 7
takes 49 tiles. Side 8 takes 64. With 50 tiles you can lay a
7 × 7 patio and have one tile left over — 50 isn't a perfect
square, so it doesn't fit a whole-square patio.
The same logic applies to anything arranged in a square: a chessboard
(8 × 8 = 64 squares), a Rubik's cube face (3 × 3 = 9), a 12 × 12
multiplication chart (144 cells). If you can lay it out in a square,
the count is a perfect square.
Worksheet
These aren't graded. Get them right, get them wrong — the goal is to
build recall for 1² through 12².
Question 1 of 3
Try it
What is 9²?
Multiple choice: what is nine squared? Four answer cards: eighteen, eighty-one, twenty-seven, seventy-two.Perfect squares
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
Going further
Perfect cubes are next. Same idea, three dimensions: 5³ = 125 is
the count of unit cubes in a 5 × 5 × 5 stack, and ³√125 = 5 is
the side length.
Square roots of non-perfect squares — like √50 — sit between two
whole numbers. √50 is between 7 and 8, closer to 7. Estimating
those values is the next step.