Area of Circles

Learn how to calculate the area of a circle using the formula A = πr².

Intermediate25 minLesson
Printable worksheet

Definition

The area of a circle is the amount of space inside the circle. To find it, we use the formula:
A=πr2A = \pi r^2
Where:
  • AA = area
  • π\pi (pi) ≈ 3.14159... (approximately 3.14)
  • rr = radius (distance from center to edge)
Key Insight: The area depends on the radius *squared*. If you double the radius, the area increases 4 times!

Try it now

What is the formula for the area of a circle?

Worked Examples

A circular pool has a radius of 5 meters. What is its area?
1

Write the formula

A=πr2A = \pi r^2 → Formula identified
2

Substitute the radius

A=π×52A = \pi \times 5^2 → A=π×25A = \pi \times 25
3

Multiply by π

A=25πA = 25\pi or A≈25×3.14A \approx 25 \times 3.14 → A≈78.5A \approx 78.5
4

Add units

Area is measured in square units → A≈78.5 m2A \approx 78.5 \text{ m}^2

Common Mistakes

Using diameter instead of radius in the formula

Why it's wrong: The formula requires radius, not diameter. Using diameter gives an answer 4 times too large!

Correct: Always check: radius = diameter ÷ 2. Then use A=πr2A = \pi r^2.

Forgetting to square the radius

Why it's wrong: Writing A=πrA = \pi r instead of A=πr2A = \pi r^2 gives a linear, not squared, relationship.

Correct: Remember: square the radius FIRST, then multiply by π. A=π×r×rA = \pi \times r \times r

Confusing area with circumference

Why it's wrong: Circumference (C=2πrC = 2\pi r) measures the perimeter. Area (A=πr2A = \pi r^2) measures the space inside.

Correct: Area uses r2r^2 (squared), circumference uses rr (not squared).

Interactive Visual

Area and perimeter

r = 5

Area78.54

A=πr2=π⋅52≈78.54A = \pi r^2 = \pi \cdot 5^2 \approx 78.54

Circumference31.42

C=2πr=2π⋅5≈31.42C = 2\pi r = 2\pi \cdot 5 \approx 31.42

Diameter10

d=2r=10d = 2r = 10

Double the radius: the circumference doubles, but the area becomes 4 times larger.

Interactive Sandbox

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the formula for the area of a circle?

Why It Matters

Circle area calculations appear everywhere in daily life:
  • Pizza: A 16-inch pizza has more than twice the area of a 12-inch pizza!
  • Gardens: Calculating how much soil or mulch you need for a circular flower bed
  • Sports: The area of a basketball hoop determines how difficult it is to score
  • Engineering: Designing circular pipes, wheels, and gears requires precise area calculations
Understanding circle area helps you make smarter decisions about space and materials!

Real World Applications

Pizza Pricing

Understanding why a larger pizza is often better value per square centimeter.

Example:

A 30 cm diameter pizza for 12 euros vs two 20 cm pizzas for 8 euros each. Which is better value?

1Try It Yourself

A pizzeria sells a small pizza (diameter 25 cm) for 10 euros and a large pizza (diameter 35 cm) for 15 euros.

Which pizza gives you more area per euro?

Step 1: Choose the right answer

Calculate the area per euro for each and pick the better deal:

Garden Planning

Calculating mulch or soil needed for circular flower beds.

Example:

A circular flower bed has radius 2 meters. At 5 euros per square meter for mulch, the total cost is π×4×5≈62.80\pi \times 4 \times 5 \approx 62.80 euros.

2Try It Yourself

You want to build a circular patio with radius 3 meters. Paving stones cost 25 euros per square meter.

How much will the paving stones cost?

Step 1: Write the mathematical expression

Calculate: Area × Cost per m²

Key Takeaways

  • 1The area of a circle is calculated using A=πr2A = \pi r^2
  • 2Always use the radius (not diameter) in the formula
  • 3Square the radius first, then multiply by π (≈ 3.14)
  • 4When doubling the radius, the area increases 4 times (because 22=42^2 = 4)
  • 5Area is measured in square units (cm², m², etc.)

Frequently Asked Questions

π (pi) is the ratio of a circle's circumference to its diameter. It appears in all circle calculations because circles have a unique curved shape. π ≈ 3.14159...
π (pi) is the ratio of a circle's circumference to its diameter. It appears in all circle calculations because circles have a unique curved shape. π ≈ 3.14159...
Divide the diameter by 2 to get the radius, then use the formula. Or use A=π(d2)2=πd24A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}.
For exact answers, leave your answer in terms of π (like 25π25\pi). For approximate answers, use your calculator's π button for precision, or 3.14 for quick estimates.

Glossary

Radius
The distance from the center of a circle to any point on its edge
Diameter
The distance across a circle through its center (diameter = 2 × radius)
Pi (π)
A mathematical constant approximately equal to 3.14159, representing the ratio of circumference to diameter
Area
The amount of space inside a two-dimensional shape, measured in square units

Formula Card

Circle Area Formula

A=πr2A = \pi r^2

Area equals pi times radius squared. Where A is the area, r is the radius, and π ≈ 3.14159. Remember: diameter = 2 × radius.

Today's challengeOne problem, three tries

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