seegongsik
Grade 6 (age 11-12)

grade 6 Area of a Circle

A circle is round, so you cannot measure its area with a ruler, but cut it into equal slices like a pizza and lay them alternately point up and point down, and the shape gets closer and closer to a rectangle. That rectangle is half the circumference, πr, long and the radius, r, tall, so the area of a circle is πr². π (pi) tells you how many times longer the circumference is than the diameter; it is about 3.14. In the first picture, add slices to watch the circle open into a rectangle; the second shows the circle filling 78.5% of the square around it.

Cutting a circle open

🍕 Thinner slices, better rectangle
①Cut the circle into equal sectors and lay them alternately point up and point down: even the starting 12 slices look a lot like a rectangle
②The thinner the slices, the flatter the bumpy top and bottom edges become, until the shape is almost exactly a rectangle
③The rectangle is r, the radius, tall and πr, half the circumference, long, because the circumference is π × diameter = 2 × π × r
④So the area is length × height = πr × r = πr²
5
12

Circle and square

↔️ The same radius
The radius you chose above is used in the picture below too. Changing the radius does not change the share of the square that the circle takes up.
💡 A circle fills 78.5% of its square
①A circle of radius r fits exactly inside a square with sides of 2r
②The square's area is 2r × 2r = 4r², and the circle's area is about 3.14 × r²
③So the circle covers 3.14 ÷ 4 = 0.785, or 78.5%, of the square. With the starting radius 5, that is 78.5 out of 100
④Double the radius and the area becomes 2 × 2 = 4 times as big

Summary

Area of a Circle
S = πr²
Area of a circle = π (about 3.14) × radius × radius
Circumference
C = 2πr = πd
Circumference = π × diameter = 2 × π × radius
Why πr²?
Base(πr) × Height(r) = πr²
The rectangle made from thin slices is half the circumference long and one radius tall
🎯 At a glance
①π: how many times longer the circumference is than the diameter, about 3.14
②Area = πr²
③Circumference = 2πr = π × diameter
④Double the radius and the area is 4 times as big
⑤Cutting a circle into thin slices and rearranging them gives a rectangle, and from it the area formula

Examples & Unit Review

Example 1
Find the area of a circle with radius 5 cm. (Use 3.14 for π.)
1
Area = π × radius × radius = 3.14 × 5 × 5.
2
3.14 × 25 = 78.5, so the area is 78.5 cm².
▸ 78.5 cm²
The first picture starts with radius 5, and its bottom line shows the same 78.5.
Example 2
Find the circumference of a circle with radius 10 cm. (Use 3.14 for π.)
1
Circumference = π × diameter, and the diameter is 10 × 2 = 20 cm.
2
3.14 × 20 = 62.8, so the circumference is 62.8 cm.
▸ 62.8 cm
Watch out: π multiplies the diameter, not the radius.
Unit Review
What is the area of a circle with radius 4 cm? (Use 3.14 for π.)
①50.24 cm²
②25.12 cm²
③12.56 cm²
④16 cm²
▸ ① 50.24 cm²
1
Area = 3.14 × 4 × 4.
2
3.14 × 16 = 50.24, so it is 50.24 cm².
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