Measuring the Universe

# Circumference

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!

# Area

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