🔵 Circles: Area
Measuring inside a circle
🎯 Introduction
Now that you know how to measure AROUND a circle, let's learn to measure INSIDE! How much pizza is in that pizza? How much grass is in that circular garden?
The area of a circle equals pi times the radius squared: A = πr²
📊 Visual Guide
💡 Key Concepts
What is Area?
Area is the amount of space INSIDE a shape. It's measured in square units (like square inches or square centimeters).
📌 Real Example:
The area of a pizza tells you how much pizza you actually get to eat. The area of a rug tells you how much floor it covers.
💭 Imagine filling the inside of a circle with tiny squares. That's what area measures!
The Area Formula: A = πr²
Area of a circle = pi times radius times radius. You MUST use the radius (not diameter), and you MUST square it!
📌 Real Example:
A pizza with radius 6 inches: A = π × 6² = π × 36 ≈ 113.04 square inches of pizza!
💭 r² means r × r. So A = π × r × r.
Why We Square the Radius
Squaring makes sense because area is two-dimensional (length × width). Even though circles are round, we're still measuring a flat space.
📌 Real Example:
If you double the radius of a circle, the area becomes 4 times bigger (not 2 times). That's because of the squaring!
💭 Imagine cutting a circle into pizza slices and rearranging them into an almost-rectangle. The 'width' is the radius, and the 'length' is half the circumference.
Using Diameter to Find Area
If you're given diameter, first find the radius by dividing by 2. Then use A = πr².
📌 Real Example:
A table with diameter 4 feet: radius = 4 ÷ 2 = 2 feet. Area = π × 2² = π × 4 ≈ 12.56 square feet.
💭 Diameter → divide by 2 → radius → square it → multiply by π → area!
📝 Step-by-Step Examples
Example 1: Finding area from radius
📋 Problem: A circular pool has a radius of 10 meters. What is its area?
Write the formula
A = πr²
Substitute the radius
A = π × 10²
Square the radius first
A = π × 100
Multiply by π (use 3.14)
A = 3.14 × 100 = 314 square meters
✅ Answer: 314 m² (or 100π m²)
Example 2: Finding area from diameter
📋 Problem: A circular hole has a diameter of 6 inches. What is its area?
Find the radius first
r = diameter ÷ 2 = 6 ÷ 2 = 3 inches
Write the area formula
A = πr²
Substitute and square
A = π × 3² = π × 9
Calculate
A = 3.14 × 9 = 28.26 square inches
✅ Answer: About 28.26 in² (or 9π in²)
Example 3: Real-world problem with subtraction
📋 Problem: A rectangular board (20 in × 40 in) has a circular hole cut out with diameter 6 inches. What is the remaining area?
Find the area of the rectangle
Rectangle: A = 20 × 40 = 800 square inches
Find the area of the circle (from Example 2)
Circle: A = 9π ≈ 28.26 square inches
Subtract circle from rectangle
Remaining = 800 - 28.26 = 771.74 square inches
✅ Answer: About 771.74 in² (or 800 - 9π in²)
Example 4: Finding radius from area
📋 Problem: A circular sign has an area of 78.5 square feet. What is its radius?
Write the formula
A = πr²
Substitute the area
78.5 = 3.14 × r²
Divide both sides by 3.14
78.5 ÷ 3.14 = r², so 25 = r²
Take the square root
r = √25 = 5 feet
✅ Answer: 5 feet
⚠️ Common Mistakes to Avoid
❌ Using diameter instead of radius in A = πr²
Why it's wrong: If you use diameter, your answer will be 4 times too big!
✅ How to avoid: ALWAYS check: did they give you radius or diameter? If diameter, divide by 2 first!
❌ Forgetting to square the radius
Why it's wrong: A = πr is WRONG for area! That's closer to circumference. Area uses r².
✅ How to avoid: Remember: Area = π times r SQUARED. Write out A = π × r × r if it helps.
❌ Squaring after multiplying by π
Why it's wrong: A = (πr)² is wrong! It's A = π × (r²). Square the radius first, then multiply by π.
✅ How to avoid: Do operations in order: 1) Square r, 2) Multiply by π.
❌ Mixing up area and circumference formulas
Why it's wrong: Area (inside) = πr². Circumference (around) = 2πr or πd. They're different!
✅ How to avoid: Area has a square (r²) because it measures square units. Circumference doesn't.
Amazing work! 🎉 You now know everything about proportional relationships AND circles! You can calculate how much fence you need for a circular garden (circumference) AND how much grass seed to buy (area)!