01 / 20

Circle Area
& Circumference

Master the two essential circle formulas: learn to compute how much space a circle covers and how far around it goes.

Pre-Algebra

What Is a Circle?

center
radius \((r)\)
diameter \(d = 2r\)

A circle is the set of all points that are exactly the same distance from a fixed center point. That distance is called the radius. The diameter stretches all the way across, passing through the center: \( d = 2r \).

Key Vocabulary

Radius
Distance from the center to any point on the circle: \( r \)
Diameter
Distance across the circle through the center: \( d = 2r \)
Circumference
The distance around the circle (its perimeter)
Area
The amount of space enclosed inside the circle

Meet the Number \( \pi \)

\(d\)
For every circle:
\( \frac{C}{d} = \pi \)
\( \pi \approx 3.14159... \)
Irrational: never terminates,
never repeats

The Circumference Formula

\(r\)
Using radius:
\( C = 2\pi r \)
Using diameter:
\( C = \pi d \)

The Area Formula

\(r\)
Area of a circle:
\( A = \pi r^2 \)
Result in square units
(cm², in², ft², ...)

Circumference vs Area

Circumference
Measures the distance around the circle.
Formula: \( C = 2\pi r \)
Units: linear (cm, m, in).
Think: wrapping a string around the edge.
Area
Measures the space inside the circle.
Formula: \( A = \pi r^2 \)
Units: square (cm², m², in²).
Think: painting the entire surface.

Step-by-Step Method

1
Identify the given measurement. Do you have the radius, diameter, circumference, or area?
2
Convert to radius if needed. If you have the diameter, divide by 2: \( r = d / 2 \).
3
Choose the right formula. Distance around → \( C = 2\pi r \). Space inside → \( A = \pi r^2 \).
4
Substitute and calculate. Plug in \( r \) and evaluate. Use \( \pi \approx 3.14 \) or leave in terms of \( \pi \).
5
Label your units. Circumference → linear units. Area → square units.

Example 1: Find the Circumference

7 cm

Given: \( r = 7 \) cm

Find: \( C \)

\( C = 2\pi r \)

\( C = 2\pi(7) \)

\( C = 14\pi \approx 43.98 \) cm

Example 2: Find the Area

5 in

Given: \( r = 5 \) in

Find: \( A \)

\( A = \pi r^2 \)

\( A = \pi(5)^2 \)

\( A = \pi(25) \)

\( A = 25\pi \approx 78.54 \) in²

Example 3: Diameter Given

12 ft

Given: \( d = 12 \) ft

Find: \( C \) and \( A \)

Step 1: \( r = 12 / 2 = 6 \) ft

\( C = 2\pi(6) = 12\pi \approx 37.70 \) ft

\( A = \pi(6)^2 = 36\pi \approx 113.10 \) ft²

Working Backwards

Sometimes you know the circumference or area and need to find the radius.

From Circumference
\( C = 2\pi r \)
Solve: \( r = \dfrac{C}{2\pi} \)
From Area
\( A = \pi r^2 \)
Solve: \( r = \sqrt{\dfrac{A}{\pi}} \)

Why Does This Matter?

Pizza
How much crust (circumference) vs how much topping surface (area)?
Fencing a Garden
Circular garden: fence length = circumference, soil needed = area.
Wheels & Tires
One rotation covers a distance equal to the tire's circumference.

Common Mistakes

✕ Using Diameter in Area
Writing \( A = \pi d^2 \) instead of \( A = \pi r^2 \). Always convert to radius first.
✕ Wrong Units
Giving circumference in cm² or area in cm. Remember: C → linear, A → square.
✕ Confusing the Formulas
Using parts of both, like writing \( 2\pi r^2 \). Remember: the square \( r^2 \) is only for Area.

Formula Summary

Circumference Area
What it measures Distance around Space inside
Formula (radius) \( C = 2\pi r \) \( A = \pi r^2 \)
Formula (diameter) \( C = \pi d \) \( A = \pi \left(\frac{d}{2}\right)^2 \)
Units Linear (cm, m, in) Square (cm², m², in²)
Solve for \( r \) \( r = \frac{C}{2\pi} \) \( r = \sqrt{\frac{A}{\pi}} \)

Question 1 of 5

A circle has a radius of 10 cm. What is its circumference? (Use \( \pi \approx 3.14 \).)

Question 2 of 5

Find the area of a circle with diameter 8 inches. Leave your answer in terms of \( \pi \).

Question 3 of 5

A circular pond has a circumference of \( 94.2 \) meters. What is the approximate radius? (Use \( \pi \approx 3.14 \).)

Question 4 of 5

A sprinkler waters a circular area with radius 6 ft. How much ground does it cover? (Use \( \pi \approx 3.14 \).)

Question 5 of 5

Which formula would you use to find how much fencing is needed to enclose a circular garden?