Area and Circumference

What You’ll Learn

In this lesson you’ll learn the formulas for circumference and area of a circle and how to use them in real situations.

The Concept

Circumference is the distance around the circle (the perimeter of the circle).

$$C = 2\pi r \quad \text{or} \quad C = \pi d$$

Area is the space inside the circle.

$$A = \pi r^2$$

Where:

• $r$ = radius
• $d$ = diameter
• $\pi \approx 3.14$ (or leave it as $\pi$ for exact answers)

Important notes:

• Circumference uses the radius or diameter (linear measurement).
• Area uses the radius squared (square units).

Worked Example

1. A circle has a radius of 5 cm.

Circumference = $2\pi \times 5 = 10\pi$ cm ≈ 31.4 cm

Area = $\pi \times 5^2 = 25\pi$ cm² ≈ 78.5 cm²

2. A circle has a diameter of 12 inches.

Radius = 6 inches

Circumference = $\pi \times 12 = 12\pi$ inches ≈ 37.7 inches

Area = $\pi \times 6^2 = 36\pi$ in² ≈ 113.1 in²

3. A circle has circumference 62.8 cm. What is the radius?

$C = 2\pi r$ → $62.8 = 2\pi r$ → $r = 62.8 \div (2\pi) \approx 10$ cm

Real-World Application

Circumference and area are used all the time:

• Buying fencing for a circular garden or yard (circumference).
• Calculating how much paint or fertilizer you need for a circular area.
• Determining the size of a round tablecloth or rug.
• Engineering: Tire size, pipe capacity, or wheel rotations (distance = circumference × number of turns).
• Cooking: How much dough you need for a round pizza or cake pan.

Example: You want to put a fence around a circular pond with radius 8 feet. Circumference tells you how much fencing to buy: $2\pi \times 8 \approx 50.3$ feet.

You’ve Got This

Remember: Circumference is around the circle (uses 2πr or πd), area is inside the circle (πr²). Use the radius for both when possible. Practice with real round objects (a plate, a wheel, or a pizza) and calculate their circumference and area. It makes the formulas feel useful and concrete.

Quiz

What is the formula for the circumference of a circle?

• A.$A = \pi r^2$
• B.$C = 2\pi r$ or $\pi d$
• C.$C = 2\pi d$
• D.$C = \pi r$

What is the formula for the area of a circle?

• A.$A = 2\pi d$
• B.$C = 2\pi r$
• C.$A = \pi d$
• D.$A = \pi r^2$

A circle has radius 7 cm. What is its area?

• A.$49\pi$ cm²
• B.$14\pi$ cm²
• C.$21\pi$ cm²
• D.$98\pi$ cm²

A circle has diameter 10 inches. What is its circumference?

• A.$5\pi$ inches
• B.$20\pi$ inches
• C.$10\pi$ inches
• D.$100\pi$ inches

A circular garden has area $36\pi$ ft². What is its radius?

• A.$18$ ft
• B.$6$ ft
• C.$12$ ft
• D.$36$ ft