Calculating a circle area by rotating its diameter

Question

I thought a circle as a set of dots .

The circumference is about 314.16 dots long and the diameter 100 dots long. I am wondering if it is possible to calculate the area of a circle by rotating its diameter.

At first I thought I could simply multiply 314.16 by 100 but I'm not getting 7853.98 (the area according to $\pi r^2$). However, it works if I divide the diameter by 4. $$ 314.16 \cdot \frac{100}{4} = 7853.98 $$ circumference * (diameter / 4) = area

Could 4 mean something I am missing ?

Answer 1

The 4 comes from two factors, each of which means you should divide by 2.

1. If you spin the whole diameter all the way around, each dot gets counted twice.
2. Only the circumferential ring has 314.16 dots, the rest all have fewer. In fact, the average number of dots is the average of the outermost ring and the innermost ring, so it should be $\frac{314.16}{2}$.

Answer 2

The area of a circle is $$A=\pi r^2=\pi\left(\frac{D}{2}\right)^2=\frac{\pi D^2}{4}$$ The circumference is given by $$C=2\pi r=\pi D$$ So the area in terms of the circumference is $$A=\frac{C^2}{4\pi}=\frac{CD}{4}$$

Answer 3

The area of a circle is $A=\pi r^2$. The circumference is $l=\pi d=2\pi r$. using $d=2 r$ you can write the area $$A=\frac{lr}2=\frac{ld}4$$

Answer 4

No, multiplication will not work, since it models square-like areas, and when you compute the product, the center ppoint is not moving, and the outside is adding more than the rest of the points. You could do it using integration as a "generalization for multiplication" -- in other words, $$ A = \int_0^r 2\pi s\ ds = \pi r^2. $$