Consider a billion-sided regular polygon that is circumscribed around a circle of radius r; how are its area and perimeter related?
same as for any regular n-gon: area = 1/2 perimeter * apothem
But it will be very close the the circle.
To understand how the area and perimeter of a billion-sided regular polygon are related, we can start by examining a simpler case, such as a regular polygon with a smaller number of sides.
Let's start with a regular polygon with, for example, 6 sides (a hexagon) circumscribed around a circle of radius r. The area of this hexagon can be found by dividing it into six congruent equilateral triangles. The formula to calculate the area of an equilateral triangle with side length 's' is:
Area = (√3/4) * s²
Since the hexagon is made up of six equilateral triangles, the side length 's' is equal to the radius of the circle, r. Therefore, the area of the hexagon can be expressed as:
Area of hexagon with radius r = (6 * (√3/4) * r²)
Similarly, the perimeter of the hexagon is simply the sum of the lengths of its six sides, which are all equal to 'r'. Therefore, the perimeter can be expressed as:
Perimeter of hexagon with radius r = 6 * r
Now, let's extend this reasoning to a regular polygon with a much larger number of sides, such as a billion-sided polygon. As the number of sides increases, the regular polygon approaches the shape of a circle. In other words, if you were to increase the number of sides to infinity, you would get a perfect circle.
In a circle, the area is given by the formula:
Area of circle with radius r = π * r²
And the circumference (perimeter) of a circle is given by:
Circumference of circle with radius r = 2 * π * r
Therefore, as the number of sides of the regular polygon increases, its area approaches the area of a circle, and its perimeter approaches the circumference of a circle. In this case, as the number of sides approaches infinity (a circle), the area and perimeter of the regular polygon will be equal to the area and circumference of the circle.
So, for a billion-sided regular polygon circumscribed around a circle of radius r, its area and perimeter would be approximately equal to the area and circumference of a circle with the same radius r:
Area ≈ π * r²
Perimeter ≈ 2 * π * r
It's important to note that the approximation becomes more accurate as the number of sides of the regular polygon increases.