Area of a Square — Complete Guide (Grade 8)
📐 What is a Square?
A square is a special quadrilateral (4-sided polygon) where:
- ✅ All four sides are equal in length
- ✅ All four interior angles are right angles (each = $90°$)
- ✅ Opposite sides are parallel
- ✅ Diagonals are equal, bisect each other at $90°$
A square is simultaneously a rectangle, a rhombus, and a parallelogram — it inherits properties from all of them!
📏 The Area Formula
$$A = s^2$$
where $s$ = length of one side of the square
Why does this work? Since a square is a rectangle with equal sides, and the area of a rectangle is $\text{length} \times \text{width}$, we get:
$$A = l \times w = s \times s = s^2$$
Alternative Formula: Using the Diagonal
If you know the diagonal $d$ of the square instead of the side, you can find the area using:
$$A = \frac{d^2}{2}$$
Derivation: By the Pythagorean theorem, the diagonal of a square with side $s$ is:
$$d = s\sqrt{2}$$
Squaring both sides: $d^2 = 2s^2$, so $s^2 = \dfrac{d^2}{2}$. Since $A = s^2$, we get $A = \dfrac{d^2}{2}$.