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Obtaining the general equation to the area of regular polygon

Introduction

In Primary School, we learned that a regular polygon can be divided into a number of congruent, equilateral triangles. This number is the same as the polygon’s number of sides. The area of the polygon can be calculated by adding up the areas of all these triangles. We hope to investigate the relationship between a polygon’s area and ...view middle of the document...

As you see, the more sides a polygon has, the closer it gets to a circle. Therefore, we decided to extend the topic to include how to calculate the area of a circle and how we could find pi. Here we show how we obtain pi with our formula step by step:

Let s be the length of the sides, k be the length of the height of the triangle and n be the number of sides of the polygon.

Step 1: Express the area of polygon with our formula

Area of polygon

= Area of Isosceles Triangle Number of sides

∵r tends to k (when n tends to ∞) (from the picture, r is approximately equal to k if both of them are from the midpoint of the figure to the edge. Therefore, the more sides a regular polygon has (the value of n is larger), the smaller the difference will be between r and k. Thus, when infinitely many triangles are drawn (when n tends to infinity), r tends to k)

Step 2: Find s in terms of r and n

Step 3: Sub into

Step 4: Compare with

tends to ∞

tends to

If we replace ∞ by a huge number, the result will be a number close to pi (3.1415926535897…). The larger the number replaced, the closer to pi it is.

Here’s an excel spreadsheet showing the theory.

Further Exploration 2

How to find the length of the sides if only the area and number of sides are given?

Our formula:

Limitations of the investigation/exploration

Although we have successfully come up with the general formula we aimed to find i.e the area of any regular polygon provided that we know the number of sides and length of one side, the formula is not capable of finding areas of non-regular polygons. However, most shapes in real life are not regular polygons so our formula could not be applied to them.

Difficulties encountered

It is uneasy for us to think...

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