Lesson 65: Circumference and Pi

New Concept

Welcome to Saxon Math 1! This course builds a strong foundation by connecting arithmetic to geometry and pre-algebra, preparing you for future mathematical challenges.

What’s next

We begin our journey with a core concept in geometry: circles. Soon, you'll dive into worked examples on calculating circumference using the special number $\pi$.

Property

The circumference of a circle is $\pi$ times the diameter of the circle. This idea is expressed by the formula $C = \pi d$. Because its value does not vary, $\pi$ is a constant.

Examples

A plate with a diameter of 10 inches has a circumference of $C = \pi \cdot 10 \text{ in} = 10\pi \text{ in}$.
A wheel with a diameter of 70 cm has a circumference of $C = \pi \cdot 70 \text{ cm} = 70\pi \text{ cm}$.

Explanation

Think of it this way: for any circle, the distance around is always a bit more than three times the distance across. That 'bit more than three' is pi ($π$), a cool, endless number that acts as a secret code for every circle in the universe!

Property

Since the diameter is always twice the radius ($d=2r$), you can find the circumference directly from the radius using the alternative formula: $C = 2\pi r$.

Examples

A circle with a radius of 5 cm has a circumference of $C = 2 \cdot \pi \cdot 5 \text{ cm} = 10\pi \text{ cm}$.
For a radius of 14 ft, use $\frac{22}{7}$ for pi: $C \approx 2 \cdot \frac{22}{7} \cdot 14 \text{ ft} = 88 \text{ ft}$.

Explanation

Don't have the full diameter? No problem! If you know the radius—the distance from the center to the edge—just double it to get the diameter, and then multiply by pi. This formula is a handy shortcut that combines both steps into one neat package.

Property

Since $\pi$ is an irrational number with a non-repeating decimal, we use approximations for calculations. The two most common approximations for $\pi$ are $3.14$ and $\frac{22}{7}$.

Examples

Diameter = 10 ft: $C \approx 3.14 \cdot 10 \text{ ft} = 31.4 \text{ ft}$.
Diameter = 21 in: $C \approx \frac{22}{7} \cdot 21 \text{ in} = 22 \cdot 3 \text{ in} = 66 \text{ in}$.

Explanation

Pi’s decimal goes on forever, which isn’t practical for most homework! So, we use $3.14$ or $\frac{22}{7}$ as 'close enough' stand-ins. Pro tip: use $\frac{22}{7}$ when your diameter or radius is a multiple of 7 to make calculations a breeze!