Circle as an infinite sided polygon

CIRCLE AS AN INFINITE SIDED POLYGON

INTRODUCTION

A circle is defined as a closed loop curve belonging to the conic family. It is a set of points in a plane that are at equal distances from a fixed point which is the center of the circle. This equal distance is called as the radius of the circle. A circle in fact is a special case of an ellipse i.e. an ellipse with equal major and minor axis is a circle or more mathematically an ellipse with zero eccentricity is a circle. In contrast a regular polygon is again a closed loop plane figure that is bounded by finite chain of straight line segments. The segments are the sides of the polygon. The number of segments depend on the type of polygon in question. For example a pentagon has 5 sides, hexagon has 6 sides and so on. This article aims to prove that a circle can be approximated as a polygon with large number of segments such that the center of gravity of circle can be determined by using the polygon and trigonometry without the need for the equation of the circle.

ASSUMPTIONS

1. The polygon considered is a regular polygon

2. The number of sides is a large finite value

CALCULATION

From the previous posts, it is evident that an ‘n’ sided regular polygon can always be divided into equal number of isosceles triangles. As the number of sides of the polygon increases, the number of triangles required to equally divide the polygons also increases. So this implies that an infinite sided polygon can be literally divided into infinite triangles. Thus the individual triangle as represented in figure 2 is very thin with the base angles approaching 90°.

Consider an infinite sided polygon. Infinite side signifies large number of sides. Let the number of sides be 1000. Hence this polygon can be equally divided into 1000 isosceles triangles. Figure 1 represents how a circle can be discretized into large sided polygon.

Fig .1 Circle as an infinite sided polygon

Fig .2 Thin isosceles triangle

The sum of angles of the polygon can be determined using the equation,

Hence individual angle (θ) of the polygon is equal to S/1000 which is 179.64°

Now the isosceles side of the triangle bisects individual angle of the polygon. Thus the individual angle (ϕ) of the triangle is half of the individual angle (θ) of the polygon.

In ∆ABC from figure 2, sum of interior angles is equal to 180°

Where, φ - top angle or the third angle of the triangle

Side BC = a - Base of triangle (m)

AM = h – Height of triangle (m)

AB = AC = b – Isosceles sides of triangle (m)

In order to prove that the polygon can be approximated as a circle, the height and isosceles sides of the triangle must be proved equal.

In ∆ABC, ⦤ABC = ⦤ACB = ϕ = 89.82°, ⦤BAC = φ = 0.36°

Consider ∆AMB,

Thus from above equation it is clearly evident that h is almost equal to b or in other words length AM is almost equal to length AB or AC. Therefore ‘h’ or ‘b’ is indistinguishable up to 5 decimal places.

Since it is proved that quantity b ≈ h, every point on the side of the polygon is located at a distance of ‘h’ from the center of polygon.

Consider the same ∆ABC inscribed inside the polygon. Point M has coordinates M ≡ (h, 0) while origin is located at O ≡ (0, 0). Since AB ≈ AM, BM ≪ AM, Point B will have the same coordinates as point M. Thus B ≡ (h, 0)

Test for Polygon to qualify as a circle

The equation of circle is,

r – Radius of the circle (m)

Since we assumed that a 1000 sided polygon must approximately represent a circle, we will solve the LHS of equation of circle to check if RHS is valid at point B

Neglecting negative value, we get radius r = h

Hence the polygon satisfies the condition of circle not only at point B but on any point on its perimeter.

INSIGHTS

1    

1    1. As the number of sides of the polygon increase, the individual angle of the polygon approach 180° but never equal to that.

2    2. The straight edges of the polygon form the curvature of the circle.

CONCLUSION

Thus an infinite sided polygon can be approximated as a circle with reasonable accuracy with the radius of circle equaling to height of the polygon. It is to be noted that in this example, a 1000 sided polygon was considered whereas a higher sided polygon would give more accuracy and thus best simulate a circle.