Square properties

A square is a four-sided shape where all four sides are equal and all four angles are right angles ($90^\circ$).
Because the sides are equal, everything about a square depends on one number: the side length $s$.

The perimeter is the distance around, so it is $4s$.
The area is the space inside, which is $s \times s = s^2$.
For example, a square with side $5$ has perimeter $20$ and area $25$.

The diagonal joins two opposite corners.
It splits the square into two right triangles with legs $s$ and $s$.
By the Pythagorean theorem, the diagonal is $\sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}$.
So a square with side $3$ has a diagonal of $3\sqrt{2}$.

A useful pattern: connect the midpoints of a square's four sides to make a new, smaller square.
The new square always has exactly half the area of the original.
Its side is the original side divided by $\sqrt{2}$.

These facts show up in area questions, in geometry with right triangles, and in patterns where a square keeps getting subdivided.
Knowing that area is $s^2$ lets you jump between a square's side and its area in one step.

A square has a perimeter of $40$. What is its area?
The perimeter is $4s = 40$, so the side is $s = 10$.
The area is $s^2 = 10^2 = 100$.

A square has a side length of $4$. How long is its diagonal?
The diagonal of a square is $s\sqrt{2}$.
So it is $4\sqrt{2}$.

A square has an area of $16$. The midpoints of its sides are joined to form a new square. What is the area of the new square?
Joining the midpoints always gives half the area.
So the new square has area $\frac{16}{2} = 8$.