Square

Square

A rectangle with all four sides of equal length. Formally, a square is a quadrilateral with four congruent sides and four congruent angles (all 90°).

Square

Area = s²

Key Formula

A = s^2 \qquad P = 4s \qquad d = s\sqrt{2}

Where:

$A$ = Area of the square
$P$ = Perimeter of the square
$s$ = Length of one side of the square
$d$ = Length of a diagonal of the square

Worked Example

Problem: A square has a side length of 6 cm. Find its area, perimeter, and diagonal length.

Step 1: Find the area using the formula A = s².

A = 6^2 = 36 \text{ cm}^2

Step 2: Find the perimeter by multiplying the side length by 4.

P = 4 \times 6 = 24 \text{ cm}

Step 3: Find the diagonal using d = s√2. Each diagonal of a square cuts it into two 45-45-90 right triangles.

d = 6\sqrt{2} \approx 8.49 \text{ cm}

Answer: The square has an area of 36 cm², a perimeter of 24 cm, and a diagonal of 6√2 ≈ 8.49 cm.

Another Example

This example works in reverse — starting from the area and finding the side length, which is a common exam question format.

Problem: The area of a square is 100 m². Find the side length, perimeter, and diagonal.

Step 1: Work backward from the area formula to find the side length. Since A = s², take the square root of both sides.

s = \sqrt{A} = \sqrt{100} = 10 \text{ m}

Step 2: Calculate the perimeter.

P = 4 \times 10 = 40 \text{ m}

Step 3: Calculate the diagonal.

d = 10\sqrt{2} \approx 14.14 \text{ m}

Answer: The side length is 10 m, the perimeter is 40 m, and the diagonal is 10√2 ≈ 14.14 m.

Frequently Asked Questions

What is the difference between a square and a rectangle?

Every square is a rectangle, but not every rectangle is a square. A rectangle requires four right angles and opposite sides that are equal. A square adds the extra condition that all four sides must be equal. So a square is a special case of a rectangle.

Is a square a rhombus?

Yes. A rhombus is a quadrilateral with four equal sides, and a square meets that requirement. The difference is that a rhombus can have angles other than 90°, while a square must have four 90° angles. A square is a rhombus with right angles.

How do you find the diagonal of a square?

Use the formula d = s√2, where s is the side length. This comes from the Pythagorean theorem: a diagonal splits the square into two right triangles, each with legs of length s. So d = √(s² + s²) = √(2s²) = s√2.

Square vs. Rectangle

	Square	Rectangle
Sides	All four sides are equal	Opposite sides are equal; adjacent sides can differ
Angles	All four angles are 90°	All four angles are 90°
Area formula	A = s²	A = l × w
Diagonals	Equal in length and bisect each other at 90°	Equal in length and bisect each other, but not necessarily at 90°
Lines of symmetry	4	2

Why It Matters

Squares appear constantly in math — from basic area and perimeter problems to coordinate geometry, tiling, and algebra (where "squaring" a number literally means finding the area of a square with that side length). Understanding squares is also essential for working with the Pythagorean theorem, since it involves squares built on the sides of a right triangle. In everyday life, squares show up in floor tiles, game boards, screens, and architectural design.

Common Mistakes

Mistake: Confusing area and perimeter formulas — for example, computing 4s when asked for area, or s² when asked for perimeter.

Correction: Remember: area (A = s²) measures the space inside and is in square units, while perimeter (P = 4s) measures the distance around and is in linear units.

Mistake: Forgetting that a square IS a rectangle and a rhombus. Students sometimes say "it's a square, not a rectangle."

Correction: A square satisfies all definitions of a rectangle, a rhombus, and a parallelogram. These categories overlap. A square is the most specific type — it belongs to all of them.

Related Terms

Rectangle — A square is a rectangle with equal sides
Rhombus — A square is a rhombus with right angles
Quadrilateral — General category: any four-sided polygon
Parallelogram — A square is a special parallelogram
Congruent — All four sides of a square are congruent
Angle — Each interior angle of a square is 90°
Side of a Polygon — A square has four equal sides
Polygon — A square is a regular four-sided polygon