Lesson 47: Circumference

New Concept

Circumference is the distance around a circle. We calculate it using its diameter or radius and the special number pi ($\pi$).

To find the circumference ($C$) of a circle, we multiply the diameter ($d$) of the circle by $\pi$.

Since a diameter is equal to two radii ($2r$), we can also use the formula:

What’s next

This is just the foundation. Next, you'll apply these formulas in worked examples and practice problems to master calculating circumference.

Property

The exact number of diameters in a circumference is represented by the Greek letter $\pi$. We use the approximation $\pi \approx 3.14$.

Examples

A plate with a circumference of 78 cm and a diameter of 25 cm has a ratio of $\frac{78}{25} \approx 3.12$.
A trash can with a circumference of 122 cm and a diameter of 38 cm has a ratio of $\frac{122}{38} \approx 3.21$.
For any perfect circle, the ratio of its circumference (C) to its diameter (d) is exactly $\pi$: $\frac{C}{d} = \pi$.

Explanation

Pi is the ultimate magic number for any circle! Imagine you measure a circle's distance around (circumference) and divide it by the distance across (diameter)—you always get pi. It's a super long, never-ending decimal, so we just call it $\pi$ and use 3.14 as a handy shortcut. It’s the secret ingredient for all circle-related math!

Property

To find the circumference of a circle, multiply the diameter of the circle by $\pi$. The formula is $C = \pi d$.

Examples

A bicycle tire with a 20-inch diameter has a circumference of $C \approx 3.14 \cdot 20 \text{ in.} = 62.8 \text{ in.}$
A penny with a 0.75-inch diameter has a circumference of $C \approx 3.14 \cdot 0.75 \text{ in.} \approx 2.36 \text{ in.}$
A plate with a diameter of 10 inches has a circumference of $C \approx 3.14 \cdot 10 \text{ in.} = 31.4 \text{ in.}$

Explanation

If you know the distance straight across a circle, you've got a golden ticket! This distance is the diameter, and all you have to do is multiply it by our good friend $\pi$ (about 3.14) to find the circumference. It’s a simple one-step trick to calculate the full distance around a pizza, a coin, or a wheel.

Property

Since a diameter is equal to two radii (2r), the formula for circumference using the radius (r) is $C = 2\pi r$.

Examples

A circle with a 2-inch radius has a circumference of $C \approx 2 \cdot 3.14 \cdot 2 \text{ in.} = 12.56 \text{ in.}$
A circle with a 3 cm radius has a circumference of $C \approx 2 \cdot 3.14 \cdot 3 \text{ cm} = 18.84 \text{ cm.}$
A circle with a 5-inch radius has a circumference of $C \approx 2 \cdot 3.14 \cdot 5 \text{ in.} = 31.4 \text{ in.}$

Explanation

Only know the radius? No problem! The radius is the distance from the center to the edge, so just double it to get the full diameter. The formula $C = 2\pi r$ does this for you by combining the doubling and the multiplication with $\pi$ into one smooth move. It's like getting a two-for-one deal for your calculation!