Square area and perimeter
A square is a rectangle where all four sides are equal. That single constraint lets you simplify the general rectangle formulas into shorter forms. Every square calculation reduces to working with one variable — the side length.
Core formulas
Area: A = s² (s times s)
Perimeter: P = 4s
where s = the length of one side
Anatomy of a square
All four sides are the same length (s). All four interior angles are exactly 90°. Opposite sides are parallel. Unlike a general rectangle, you only need one measurement — the side length — to describe a square completely.
The diagonal of a square can be found with the Pythagorean theorem: d = s√2, though that is not needed for area and perimeter questions.
Four question types
Because you only have one variable, every square question is one of these four types.
Find area (know side)
A = s × s = s²
Find perimeter (know side)
P = 4 × s
Find side (know area)
s = √A
Find side (know perimeter)
s = P ÷ 4
Why s² and not s × 2?
A common confusion: A = s² means s squared (s multiplied by itself), not s times 2. This matters enormously.
Correct: s = 6
A = 6² = 6 × 6 = 36
Wrong (common mistake): s = 6
A = 6 × 2 = 12 ← this is not the area formula
The exponent ² always means “multiply the base by itself,” never “multiply the base by 2.”
Worked examples
Example 1: find the area
A square has a side length of 9 cm. Find the area.
A = s² = 9² = 9 × 9 = 81 cm²
Square the side length. 9 times 9 is 81. Attach the square unit.
Example 2: find the perimeter
A square has a side length of 7 m. Find the perimeter.
P = 4 × s = 4 × 7 = 28 m
All four sides are identical, so multiply the side by 4. The unit stays linear.
Example 3: find the side from the area
A square has an area of 144 ft². Find the side length.
A = s² → s = √A = √144 = 12 ft
Reverse the squaring operation by taking the square root. Only the positive root makes sense as a side length. For the practice widget, only perfect-square areas appear so the root is always a whole number.
Example 4: find the side from the perimeter
A square has a perimeter of 36 in. Find the side length.
P = 4s → s = P ÷ 4 = 36 ÷ 4 = 9 in
Because all four sides are equal, dividing the perimeter by 4 always gives the side length directly.
Related shape: the rectangle
Every square is a rectangle, but not every rectangle is a square. The rectangle formulas become the square formulas whenever l = w.
Square (all sides equal)
A = s²
P = 4s
Rectangle (general case)
A = l × w
P = 2(l + w)
Refreshable sample
Try one square question
Work through the problem before checking. The widget generates a different question each time you refresh.
Common mistakes to avoid
- Confusing s² with 2s. s² means s × s. Writing 2 × s gives the wrong area. Always multiply the side by itself.
- Using the rectangle perimeter formula. P = 2(l + w) works when l ≠ w. For a square, use P = 4s directly — it is simpler and less error-prone.
- Forgetting the square root when working backward from area. If A = 100, then s = √100 = 10, not s = 100 ÷ something. The inverse of squaring is the square root.
- Mixing up which formula to apply. Know whether the problem gives you a side, an area, or a perimeter before you start calculating.
- Wrong units. Area is in square units (m², ft²). Perimeter and side length are in linear units (m, ft). Match the unit to the type of measurement.