Question

prove area of rectangle

Answer 1

Theorem: The area of a rectangle is the product of its length and width.

Consider the square below with side length  units.  The square is divided into four parts: two squares and two rectangles. We already know that the area of the two squares are  and We do not know the area of the rectangle yet because that is what we are trying to prove.


Now, let be the area of each rectangle shown above. Clearly, the area of the largest square is the sum of the areas of the two smaller squares and the two rectangles. In equation form, we have

$(x + y)(x + y) = x ^{2} + y ^{2} + 2a$
Expanding the left hand side, we have

$x ^{2}2xy + y ^{2} = x ^{2} + y ^{2} + 2a$

Subtracting x^2 + y^2 from both sides results to

$2xy = 2a$
Solving for gives us

$a =xy$

But  x and y  are the length and width of the rectangle, therefore, the area of any rectangle is the product of its length and its width.