知识点 4

Solve problems involving arcs, arc length, and sector area.

Arc length and sector area are fractions of a full circle. Match the formula to degrees or radians before you calculate.

核心知识

Arc length and sector area are fractions of the full circumference and area: multiply by 𝜃360° (degrees) or 𝜃2𝜋 (radians).

深入理解

Rule: Arc length and sector area are fractions of the full circle.

  • Use 𝜃360∘ when the angle is in degrees.
  • Use 𝑟𝜃 and 12𝑟2𝜃 when the angle is in radians.
  • Arc length is a distance; sector area is a region.

Choose the formula that matches what the question asks for.

分步讲解

  1. Identify the central angle 𝜃 and the radius 𝑟.
  2. Determine the fraction of the circle: 𝜃360° (or 𝜃2𝜋 in radians).
  3. For arc length: multiply the fraction by 2𝜋𝑟.
  4. For sector area: multiply the fraction by 𝜋𝑟2.

常见误解

  • Using degrees in the radian formula (or vice versa).
  • Computing the area of the full circle instead of the sector.
  • Confusing arc length (a distance) with sector area (a region).
题目

示例解析

A circle has a radius of 6. What is the area of a sector with a central angle of 120°?

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