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.
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| Fig .1 Circle as an infinite sided polygon |
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| 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.
