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When you find the number close to the area of a circle, where you keep putting more and more sided shapes on the inside and outside of the circle, can you just do that with triangles and find the average of the two numbers and get the area of the circle?
Answer 1:

As you probably know, the area of a circle is pi (r2) where pi = 3.14159265... and r2 = r r where r is the radius (half the diameter) of the circle. As you mention, if you inscribe a regular polygon inside a circle and circumscribe the circle with a similar regular polygon with the same number of sides and take the average of the areas of those two polygons you get a number that is close to the area of the circle.
As you would expect, the as the number of sides of the polygons gets large, the average of the two areas gets closer and closer to the area of the circle.

Since most people don't learn the math required to solve this problem until some time in high school, I'll give a few examples first.
First, lets say n equals the number of sides of the polygons used: n = 3 for triangles, n = 4 for squares, n = 5 for pentagons, etc. According to the formula I came up with, here is the average of the two areas (compare to pi r2):
n = 3:
3.2476 r2
n = 4:
3 r2
n = 5:
3.0052 r2
n = 6:
3.0311 r2
n = 10:
3.0941 r2
n = 100:
3.1411 r2
n = 1000:
3.14159 r2


As you can see, the number before r2 gets closer and closer to pi the higher n is.
Using just triangles or squares isn't actually that bad of an approximation as it turns out.

For those who have had trigonometry (or have a scientific calculator to play with), here is the formula I got for the average of the areas of two regular polygons with n sides, one that is inscribed in a circle of radius r, the other circumscribing the circle:

0.5 n [ tan(pi/n) + sin (pi/n) cos (pi/n) ] r2

Answer 2:

Thank you for your interesting question. Its a classical way to estimate the area of a circle by the procedure you describe.

You can also try out this procedure by drawing a fairly large circle and then draw inner triangles and corresponding outer triangles. Its useful to mark the center of the circle,then draw diameters and connect them at the boundary of the circle.
By successively increasing the number of triangles and compute the inner and outer areas and the average of them you will see that the computed areas will approach a certain value, the area of the triangle. Do you know how to compute the area of a triangle?

If you then divide the area by the radius (half the diameter) twice you will obtain a famous number. Which number?
If you have any questions just ask again.


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