Using only this photo, could you find the circumference of this carousel to within a few feet?

First we could take the height of a person who looks to be around 6 feet and figure out the radius of the circle:

5 * 6 = roughly 30 feet

With the radius we can use π to determine the circumference directly:

\large{\pi} = \dfrac{\text{circumference}}{\text{diameter}}

\large{\pi} = \dfrac{\text{circumference}}{2 \times r}

\text{circumference} = 2 \times \pi \times r

\text{circumference} \approx 2 \times 3.14 \times 30

\text{circumference} \approx 188 \text{ feet}

We correctly find that it’s approximately 188 feet around the perimeter of the wheel, all thanks to π!


How can we estimate the area of any circle given its radius?

Let’s approximate the answer using a pizza slice analogy. Below is an interactive illustration; click the arrows to change the number of slices in the pie:

As we increase n, the perimeter of the pizza approaches the circumference of the circle. As we increase the number of slices:

  • the height of the slice approaches the radius of the circle ( h = r )
  • the area of the pizza approaches the area of the circle ( area of pizza = area of circle )

This leads to a well-known equation which relates the area and radius of any circle:

\text{area of circle} = \pi r^2