Teach: Sectors ② Pi and the Circumference and Area of a Circle

Sectors ② Pi and the Circumference and Area of a Circle

Before moving on to the formulas for sectors, let us review the formulas for the circumference and area of a circle and pi π. Once these are understood, sectors can be calculated easily as “parts of a circle.”

Pi π

For every circle, the value of circumference ÷ diameter is always constant, approximately 3.14159…. This constant value is called pi and is represented by the Greek letter π.
　π ＝ circumference / diameter ≒ 3.14

From middle school onward, calculations often leave pi as π. Remember, “do not change π to 3.14” (use 3.14 only when specifically instructed).

Circumference

For a circle with radius r and diameter d ＝ 2r, its circumference ℓ is:
　ℓ ＝ 2πr　(from the radius)
　ℓ ＝ πd　(from the diameter)

Area of a Circle

The area S of a circle with radius r is:
　S ＝ πr²

If π is taken as 3.14, the area of a circle with radius 5 cm is 3.14 × 25 ＝ 78.5 cm². If π is left as it is, write 25π cm².

Example ①
For a circle with radius 8 cm, find the following. Leave π as it is.
(1) Circumference　(2) Area
Solution:
(1) ℓ = 2π × 8 = 16π (cm)
(2) S = π × 8² = 64π (cm²)

Calculations Using the Diameter

When the diameter is given, first convert it to the radius (÷2), or use ℓ = πd directly.
　Example: Circle with diameter 10 cm → circumference = π×10 = 10π cm, area = π×5² = 25π cm².

Example ②
Find the circumference and area of a circle with diameter 14 cm. Leave π as it is.
Solution:
Radius = 14÷2 = 7 cm.
Circumference = 2π × 7 = 14π (cm) (or πd = 14π is also acceptable)
Area = π × 7² = 49π (cm²)
Key Points
  • Pi π ≒ 3.14159…. In middle school, leave it as π unless instructed otherwise.
  • Circumference = 2πr = πd
  • Area of a circle = πr²
  • For a problem beginning with the diameter, first convert it to the radius.

Practice Problems

  1. Find the circumference and area of a circle with radius 6 cm, leaving π in the answers.
  2. Find the circumference and area of a circle with diameter 20 cm, leaving π in the answers.
  3. A circle has circumference 12π cm. Find its radius and area, leaving π as it is.

Answers and Explanations

  1. Answer: Circumference 12π cm, area 36π cm²
    Explanation: ℓ = 2π×6 = 12π, S = π×6² = 36π.
  2. Answer: Circumference 20π cm, area 100π cm²
    Explanation: Radius = 10. ℓ = 2π×10 = 20π, S = π×10² = 100π.
  3. Answer: Radius 6 cm, area 36π cm²
    Explanation: 2πr = 12π → r = 6. S = π×6² = 36π.

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