Friday, October 30, 2015

Unizor - Geometry2D - Area of a Circle

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Area of a Circle

1. Definition

No matter how small our area measurement unit (a square with a side of 1) is, we will always have problems to measure exactly the area of a circle of a radius 1.
So, we have no choice, but to resort to a process that leads to an area of a circle as its limit.
This process might involve filling a circle with as many as possible squares of a side 1, then choose a smaller unit square of a side size, say, 1/10 and fill the empty area and continue this process getting more and more precise value with smaller and smaller measurement units.

In this lecture I'd like to introduce a different method that leads to a formula for the area of a circle more directly through another process.

Let's inscribe a regular polygon into a circle. We can start with a hexagon, similarly to a process we used to evaluate a circumference of a circle, or any other regular polygon. Next, we will double the number of edges of this polygon on every step by constructing perpendiculars to all edges and taking new vertices at points of intersection of these perpendiculars with a circle.

Now we state without rigorous proof that inasmuch as the perimeter of these polygons tends to a circumference of a circle, their area tends to some limit that can be used as a definition of the area of a circle.

Moreover, it can be proven that the limit of the area of polygons exists and is always the same, regardless of which polygon we start with and how we increase the number of its vertices, as long as the longest edge of polygons decreases to zero.

With this theorem we can state that the definition of an area of a circle is correct. It exists as a limit of areas of polygons transforming during a process described above and is unique.

2. Formula for Area of a Circle

Our next task is to evaluate the limit of the area of these polygons. That limit will be the area of a circle.

Let's examine a polygon with N sides obtained at some point during our process of increasing the number of vertices. Connect all vertices of our N-sided polygon with a center of a circle and consider the area of a polygon as a sum of areas of all N triangles with one vertex at the center of a circle and two other vertices being adjacent vertices of a polygon.
All these triangles are congruent and the area of each of them equals to
where a is the length of the edge of a polygon serving as an edge of a triangle and h - the altitude of each triangle dropped from the center of a circle onto an opposite edge.

Since we have N such triangles in a polygon, the total area of a polygon equals to
Now notice that N·a is a perimeter of our polygon p. So, we can say that the area of a polygon equals to

Now it's time to observe the trend of this area. As the number of vertices increases to infinity, the perimeter of inscribed polygons tends to a circumference of a circle that is equal to 2πR, where R is the radius of a circle.
At the same time the altitude h tends to be closer and closer to a radius of a circle.

So, as we approach the limit, the area of a circle will be equal to
2πR·R/2 = πR²

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