Circles: Area

Free sample questions, a clear explanation, and 5 practice skills with an AI tutor that guides without giving the answer away.

Concept Review

The Area of a Circle: Pizza Math That Actually Matters

Imagine you're ordering pizza for a party. Two pizzas cost the same: one has a 12-inch diameter, the other has a 10-inch diameter. Which gives you more pizza for your money? The answer lies in understanding the area of a circle — and it might surprise you.

The Circle Area Formula

The area of any circle is calculated using the formula A = πr², where r is the radius (the distance from the center to the edge). This formula works whether you're measuring pizza, bicycle wheels, or crop circles.

But what if you only know the diameter? No problem — the radius is always exactly half the diameter. So if your pizza has a 12-inch diameter, the radius is 6 inches.

Let's Solve the Pizza Problem

The 12-inch pizza has a radius of 6 inches:

A = πr² = π × 6² = π × 36 ≈ 113.1 square inches

The 10-inch pizza has a radius of 5 inches:

A = πr² = π × 5² = π × 25 ≈ 78.5 square inches

🤯 Mind-Bending Truth

When you increase a circle's radius by just 20% (from 5 to 6 inches), the area doesn't increase by 20% — it increases by 44%!

This happens because area involves squaring the radius. Small changes in radius create surprisingly large changes in area. That 12-inch pizza gives you almost 1.5 times more food than the 10-inch pizza.

Working with Different Measurements

Whether you start with radius or diameter, the process is the same:

Step 1:Find the radius (divide diameter by 2 if needed)
Step 2:Square the radius (multiply it by itself)
Step 3:Multiply by π (≈ 3.14159)

For a garden sprinkler that waters in a circle with a 15-foot diameter, the radius is 7.5 feet. The watered area is π × 7.5² = π × 56.25 ≈ 176.7 square feet.

🔑 Key Takeaway

Next time you're choosing between different-sized circular objects — pizza, trampolines, or swimming pools — remember that small differences in radius create big differences in area. That "slightly larger" option might give you way more bang for your buck than you think.

Sample questions

1. Find the area of a circle with radius 4 cm. Use π ≈ 3.14.

○ 50.24 cm²

○ 25.12 cm²

○ 50.24 cm²

✓ 50.24 cm²? A = πr² = 3.14 × 4² = 3.14 × 16 = 50.24 cm²

Answer: 50.24 cm²? A = πr² = 3.14 × 4² = 3.14 × 16 = 50.24 cm² — Area = π × r² = 3.14 × 16 = 50.24 cm².

2. A circle has diameter 10 inches. What is its area in terms of π?

○ 25π in²

○ 100π in²

○ 10π in²

✓ 25π in²? radius = 5, so A = π × 5² = 25π

○ 25π in²

Answer: 25π in²? radius = 5, so A = π × 5² = 25π — Radius = 5 in, so area = π × 5² = 25π in².

3. Calculate the area of a circle with radius 3.5 cm. Use π ≈ 22/7.

✓ 38.5 cm²? A = (22/7) × (3.5)² = (22/7) × 12.25 = 22 × 1.75 = 38.5 cm²

○ 38.5 cm²

○ 77 cm²

○ 38.5 cm²

Answer: 38.5 cm²? A = (22/7) × (3.5)² = (22/7) × 12.25 = 22 × 1.75 = 38.5 cm² — 3.5² = 12.25, times 22/7 = 38.5 cm².

Skills in this topic

Calculate the area of a circle given the radius or diameter
Find the radius or diameter of a circle given the area
Calculate the area of a semicircle or quarter circle
Understand the mathematical relationship between a circle's area and circumference
Solve real-world problems involving the area of circular spaces

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