Area Of Rectangle

Theorem: Area of a rectangle is equal to the product of any of its two adjecent sides.

Prerequisites:
Area (definition)

Proof:

Let the given rectangle ABCD having sides equal to '$a$' and '$b$' be divided into small squares having unit area. Then the total area is equal to the total number of these squares.

If both the sides lengths are integral:

Total number of squares = $ab$
$\therefore\quad\;\;\text{Area} = ab$

If the side lengths are rational:

Let $a = p/q$ and b is integral. Then place $q$ such rectangles together, so that the total length becomes equal to $p$.

Then by previous case,
total area of $q$ rectangles $= pb$
Hence, area of $1$ rectangle $= \dfrac{pb}{q}\\
= ab$

If atleast 1 side length is irrational:

Every irrational number is surrounded by infinite rational numbers. Since area by definition is the region occupied, which is a continuous quantity, hence by continuity requirements, since every irrational number is surrounded by infinite number of rational numbers. therefore as the given equality holds for the rational side lengths so it should also hold for the irrational side lengths.

Hence the result.


Recommended:
Area of triangle
Area of rhombus
Diagonal property of rectangle