Finding circumference without using $\pi$

Solution 1:

Consider a $n$ sided regular polygon as shown in the figure below. enter image description here

Let the diameter of the polygon as shown be $d$ and the side of the polygon be $a$. Then the area of the triangle as shown in the figure is $$T = \dfrac12 \times a \times \dfrac{d}2 = \dfrac{ad}4$$ The perimeter of the polygon is $$P = na,$$ while the area of the polygon is $$A = nT = \dfrac{nad}4.$$ Hence, we get that $$A = \dfrac{Pd}4$$ Letting the number of sides $n$ tend to infinity, the polygon "tends" to a circle and we get that $$\text{Area of the circle} = \dfrac{\text{Circumference }\times \text{ diameter}}4$$ or to put it the other way around $$\text{Circumference} = \dfrac{4 \times \text{Area of the circle}}{\text{ diameter}}$$ As you see from this, there is no need for $\pi$ anywhere.

Solution 2:

Recall: $$A = \pi \left(\frac{D}{2}\right)^2$$ $$C = \pi D$$ Where $C$ is circumference, $A$ is area, and $D$ is diameter.

Thus: $$\frac{A}{C} = \frac{\pi \left(\frac{D}{2}\right)^2}{\pi D}$$ The $\pi$'s cancel out, as well as one of the D's: $$\frac{A}{C} = \frac{D}{4}$$

Solving for C: $$C = \frac{4A}{D}$$

Solution 3:

Let's write down all our formulas:

$$A = \pi r^2 = 254.34$$ $$C = 2\pi r$$ $$d = 2r = 18$$

where $A$ is area, $r$ is radius, $C$ is circumference, and $d$ is diameter. We're trying to find the expression in the second; if we look at the first formula it's almost there, but it's missing a factor of 2 (easy to solve) and there's an extra factor of $r$. But if we divide the expression in the first formula by the expression in the third formula, we get

$$\frac{254.34}{18} = A/d = \frac{\pi r}{2}$$

Almost there! Now we just need to multiply by 4:

$$2 \pi r = 4\frac{\pi r}{2} = 4A/d = 4*\frac{254.43}{18} = 56.52.$$