Area and Perimeter of a Square: Formulas and Examples

For a square with side length a, the two main formulas are:

• Area: A = a2
• Perimeter: P = 4a

Area measures the surface inside the square and uses square units, such as cm² or m². Perimeter measures the distance around the outside and uses ordinary length units, such as cm or m.

What is a square?

A square is a quadrilateral with four equal sides and four right angles. Its opposite sides are parallel. Its two diagonals are equal in length, cross at their midpoints, are perpendicular to each other, and bisect the corner angles.

A square is both a rectangle with four equal sides and a rhombus with four right angles.

Formula for the area of a square

Multiply the side length by itself:

A = a × a = a2

Here, A is the area and a is the length of one side. All lengths must use the same unit before you multiply.

Example: A square has a side length of 6 cm.

A = 6 × 6 = 36 cm2

The answer is 36 cm², not 36 cm, because area is two-dimensional.

Why the formula works

A square is a rectangle whose length and width are both a. The rectangle formula length × width therefore becomes a × a. You can also divide the square into two congruent right triangles; their combined area is still a2.

The same idea works when a square is partitioned into rectangles: add the areas of the non-overlapping parts.

Formula for the perimeter of a square

The perimeter is the sum of all four equal sides:

P = a + a + a + a = 4a

Example: If a = 6 cm, then:

P = 4 × 6 = 24 cm

Find a missing measurement

Known value	Side length	Area	Perimeter
Side a	a	a2	4a
Perimeter P	P ÷ 4	(P ÷ 4)2	P
Area A	√A	A	4√A
Diagonal d	d ÷ √2	d2 ÷ 2	2√2d

Worked examples

1. Find area from a side

A square has side length 4 cm.

A = 42 = 16 cm2

2. Find area from perimeter

A square has perimeter 28 cm. First find one side:

a = 28 ÷ 4 = 7 cm

Then calculate the area:

A = 72 = 49 cm2

3. Extend a square into a rectangle

One side of a square is extended by 5 cm, producing a rectangle with perimeter 110 cm. Let the original side be a. The new dimensions are a and a + 5.

2a + 2(a + 5) = 110
4a + 10 = 110
a = 25 cm

The rectangle is 25 cm by 30 cm, so its area is:

25 × 30 = 750 cm2

4. Path inside a square garden

A square garden is 20 m on each side. A 2 m-wide path runs around the inside edge. The remaining central square loses 2 m on both ends of each dimension:

20 - 2 - 2 = 16 m

Cultivated area: 162 = 256 m2.

5. Find perimeter from area

A square has area 36 cm². Its side is √36 = 6 cm, so its perimeter is 4 × 6 = 24 cm.

Common mistakes

• Using 4a for area or a2 for perimeter.
• Writing a square unit for perimeter or a linear unit for area.
• Mixing units, such as multiplying centimeters by meters, without converting first.
• Dividing an area by 4 to find the side; use the square root instead.
• Forgetting that a path around the inside reduces both ends of each dimension.

A quick check helps: if the side length doubles, the perimeter doubles, but the area becomes four times as large.