Proofs of Area Theorems

In this article, we will explore the proofs of area theorems, especially when dealing with parallelograms and triangles. Understanding these proofs will enhance your understanding of geometry and the properties that govern these shapes. We will see how their areas are calculated and the logic behind these calculations.

Area of parallelogram

A parallelogram is a four-sided shape in which the opposite sides are parallel and equal in length. To find its area, the following formula is used:

Area = Base × Height

Here, "base" refers to the length of the bottom side of the parallelogram, and "height" refers to the perpendicular distance between the base and the opposite side. Let's look at the proof of why this formula works.

Visual example

Consider the standard parallelogram given below:

Proving the area formula

To understand this better, consider cutting a parallelogram along one of its diagonals to form two triangles. These two triangles are similar - they have exactly the same size and shape. This similarity helps us understand that the area of the parallelogram can be seen as the sum of the areas of these two triangles.

Let's move one of these triangles to form a rectangle in which the base and height of the parallelogram are unchanged. Rectangles have a straightforward area formula: Area = Length × Width. In the rearranged rectangle, the base of the parallelogram serves as the length, and the height of the parallelogram becomes the width. Thus, the formula Area = Base × Height emerges naturally and is proven by rearranging the figure.

Area of a triangle

A triangle is a polygon with three sides. The formula for finding the area of a triangle is different from the formula for finding the area of a parallelogram. The area of a triangle is given by the formula:

Area = 1/2 × Base × Height

Visual example

Consider a triangle as shown below:

Proving the area formula

The formula for the area of a triangle can be derived by comparing it to the area of a parallelogram. Consider a triangle with base b and height h. If you create a second similar triangle and connect it with the first one, they form a parallelogram. The area of this parallelogram will be Base × Height, which is b × h.

Since the original figure was just a triangle, the area of the triangle is exactly half the area of the parallelogram. Therefore, the area of the triangle can be represented as:

Area of Triangle = 1/2 × Base × Height

Use of Heron's formula

Sometimes, you are not given the height directly, or it is cumbersome to calculate it. In such cases, Heron's formula can be useful for finding the area of a triangle when you know the lengths of all three sides.

If the lengths of the sides of a triangle are a, b and c, then the area A can be found as follows:

s = (a + b + c) / 2
Area = √(s * (s - a) * (s - b) * (s - c))

Let's consider an explicit example using Heron's formula. Let's say there is a triangle with side lengths a = 5, b = 6 and c = 7 First calculate s:

s = (5 + 6 + 7) / 2 = 9

Then, calculate the area:

Area = √(9 * (9 - 5) * (9 - 6) * (9 - 7))
Area = √(9 * 4 * 3 * 2)
Area = √(216)
Area ≈ 14.7

Conclusion

Understanding the proofs behind these area theorems strengthens your understanding of geometry. Knowing how the area formulas for parallelograms and triangles are derived from fundamental geometric principles allows you to seriously apply these concepts. Whether using the base-height relationship or applying Heron's formula, these proofs provide a strong framework for calculating the area of these common shapes.

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