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.

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.

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.