Formula

Perimeter and area of a square

Mensuration • Plane Figures • Quadrilateral

Formula

P = 4 a, A = a^{2}

A square of side

a

has perimeter

4 a

and area

a^{2}

; eliminating

a

gives area

= \frac{P ^{2}}{16}

in terms of the perimeter

P

How to use it

1To compare a square with another figure of the same perimeter, express both areas through $P$ and divide.
2The diagonal is $a 2$ , which is what relates a square to a circle it is inscribed in.
3A square of given perimeter has the largest area of any rectangle with that perimeter.

A square field has perimeter

72

m. Find its area.

Step-by-step

1
$4 a = 72 \Rightarrow a = 18$ m
Why: Recover the side from the perimeter before squaring.
2
$A = 1 8^{2} = 324$ m $^{2}$

Answer

324

^{2}

Watch out

Perimeter is linear in

a

and area quadratic, so two squares whose perimeters are in the ratio

1 : 2

have areas in the ratio

1 : 4

. Squaring the perimeter instead of the side overstates the area sixteenfold.

Also known as

area of a square, perimeter of a square, side squared, 4a