A square has side length 12 m
- Area = side × side = 12 × 12 = 144 m².
- The perimeter is 4 × 12 = 48 m; it is a different measurement.
Squares & square area
Connect an exponent to an actual square.
Squaring multiplies a number by itself
Connect side length, exponent notation and square area, and distinguish squaring from doubling.
The expression 6² means 6 × 6, not 6 × 2. A square with side length 6 cm has area 36 cm². Six rows of six unit squares connect the multiplication to the area. The side length is a linear measure; the area counts square units.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
A perfect square is a whole-number square
The non-negative integers 0, 1, 4, 9, 16, 25 and 36 are perfect squares. They can be written as n² for a whole number n. Consecutive square numbers grow by successive odd amounts: from 4² to 5², add 9; from 5² to 6², add 11. An extra row and column explain this growth.
Read the base before applying the exponent
In (−3)², the brackets make −3 the base, so the value is 9. In −3², standard order of operations squares 3 first and then takes its opposite, giving −9. A geometric side length is non-negative even though a negative number can be squared.
Worked examples
Compare (−5)² and −5²
- (−5)² = (−5) × (−5) = 25.
- −5² = −(5 × 5) = −25.
- The brackets determine which number is the base.
Guided Practice
A square has side length 9 mm. Find its area and perimeter.
A first hint
Use side × side for area and four sides for perimeter.
Compare your reasoning
Area = 81 mm². Perimeter = 36 mm. The units distinguish the two quantities.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
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Connections
Explain why (n + 1)² − n² = 2n + 1 using an added row and column. Why is their shared corner counted only once?
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