Derivation of the Proof of the Area of a Rectangle

In getting the areas of figures, calculating the area of a square is probably the easiest. We can easily calculate the area of a square by squaring its side length . For example, a square with side length 5 units has area 25 square units. What about the area of a rectangle?

Based on the finding the area of a square, our intuition tells us that it is the product of the length and the width. However, how do we prove that it is indeed true? In this post, we are going to discuss the proof of the area of a rectangle. We show that it is the product of its length and its width.

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

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

The Derivation of the Area of a Circle

Finding the area of a circle is not as easy as finding the areas of triangles, parallelograms, and trapezoids. The circumference of a circle is curved, and it is not easy to fit squares in the bounded region. Knowledge in calculus is required to prove the circle’s area formally, but in this post, I will discuss a more intuitive way of understanding why the area of a circle with radius $latex r$ is $latex \pi r^2$.

If we divide the circle into congruent sectors, we can  form the figure above. In the figure, the circle is divided into 6 equal sectors which means that each of the central angles is equal to $latex 60 ^\circ$.  Continue reading

Using the Area of a Rectangle to Derive the Area of a Triangle

The area of a rectangle is the product of its base and its height. We will use this fact to derive the area of a triangle. If we can show that the area of the rectangle and the area of a triangle are related, then we can find the formula.

To show the relationship between the two figures, we need to show that for any triangle, we can always create a rectangle using one of its side as the triangle’s base. We have four cases: acute, right, obtuse and equiangular triangles. Continue reading