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Grade 6 (age 11-12)

Area of a Circle

Area of a Circle

A circle is round, so its area is hard to measure directly, but slice it like a pizza and fit the pieces together and the shape approaches a rectangle. That rectangle has a base of half the circumference and a height of the radius, so the area is radius times radius times pi. Here you add slices to watch the circle open into a rectangle and see it fills about 78.5% of the surrounding square.

Cut a Circle and Spread It Out
🍕 Slice a Pizza → Becomes a Rectangle!
①Cut the circle into equal pizza slices
②Alternate top/bottom → looks like a rectangle!
③More slices → closer to a true rectangle
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Circle vs Square
💡 Circle ≈ 78.5% of Square
①Square with side 2r contains the circle
②Area = πr² ≈ 3.14 × r²
③About 78.5% of the square area 4r²
Summary
Area of a Circle
S = πr²
Area = π (≈ 3.14) × radius × radius
Circumference
C = 2πr = πd
Circumference = 2 × π × radius = π × diameter
Why πr²?
Base(πr) × Height(r) = πr²
Slice into sectors → rearrange to rectangle: base = half circumference, height = radius
🎯 Key Points
①π ≈ 3.14 (ratio of circumference to diameter)
②Area = πr² (proportional to radius²)
③Circumference = 2πr (proportional to radius)
④Double the radius → 4× the area
⑤Slicing and rearranging derives the formula!
Examples & Unit Review
Example 1
Find the area of a circle with radius 5 cm. (Use π = 3.14)
1
Area = π × radius × radius = 3.14 × 5 × 5.
2
3.14 × 25 = 78.5, so 78.5 cm².
78.5 cm²
The area of a circle multiplies the radius twice and then by π.
Example 2
Find the circumference of a circle with radius 10 cm. (Use π = 3.14)
1
Circumference = 2 × π × radius = 2 × 3.14 × 10.
2
6.28 × 10 = 62.8, so 62.8 cm.
62.8 cm
Circumference is the diameter (twice the radius) times π.
Unit Review
What is the area of a circle with radius 4 cm? (Use π = 3.14)
50.24 cm²
25.12 cm²
12.56 cm²
16 cm²
① 50.24 cm²
1
Area = π × radius × radius = 3.14 × 4 × 4.
2
3.14 × 16 = 50.24, so 50.24 cm².