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Arc Length and Area of a Sector

The Constant

The ratio of the circumference of a circle to its diameter, known as the constant , is always the same. In elementary school, is often approximated as . However, its actual value is:
This value continues indefinitely as a non-repeating decimal. The symbol is used to represent this constant, and it is pronounced as “pi.”
The formulas for the circumference and area of a circle are:
  • Circumference of a circle = (Diameter) (Constant )
  • Area of a circle = (Radius Radius) (Constant )
If the radius of the circle is , then the circumference and area are expressed as:

Arc Length and Area of a Sector

For a sector with radius and central angle , the arc length and area are proportional to the central angle. These are given by the following formulas:
  • If the radius is and the central angle is , the arc length and area are related as follows:
Thus, from , we derive , which leads to the formula:
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