Question

Prove that the area of a rectangle is length × breadth ​

Answer 1

Answer:

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

Consider the square below with side length x + y units. The square is divided into four parts: two squares and two rectangles. We already know that the area of the two squares are x^2 and y^2. We do not know the area of the rectangle yet because that is what we are trying to prove.

Derivation of the Proof of the Area of a Rectangle

Now, let A 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 A 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.