Arcs and Sectors

Understanding Arcs and Sectors

  • An arc is a part of the circumference (boundary) of a circle.
  • A sector is a region of a circle, enclosed by two radii and the arc between them.
  • These two parts are keystones in circular geometry as they enable us to calculate lengths and areas related to part of a circle, not just the whole circle.

Calculating with Arcs and Sectors

  • The length of an arc can be calculated using the ratio of the angle subtended by the arc to the total angle of the circle (360°), multiplied by the circumference of the circle. The formula is: Arc Length = (θ/360) x 2πr, where θ is the central angle in degrees and r is the radius of the circle.
  • The area of a sector can similarly be found using the ratio of the angle to the total angle of the circle, multiplied by the total area of the circle. The formula is: Sector Area = (θ/360) x πr².

Applying Arc and Sector Formulas

  • If given a sector, you can calculate the area of the sector and the length of the arc using the formulas above, provided you know the length of the radius and the size of the angle forming the sector.
  • Although these formulas may initially seem tricky, with enough practise they become second nature.

Arcs, Sectors, and Real-World Applications

  • In the real world, arcs and sectors are found in a large number of structures and mechanisms involving circular movement or design. Think of the angles made by the minute and hour hands on clocks, sectors in a pie chart, swings of a pendulum, field of vision, and path of sporting equipments in games like football or golf etc. where you need to consider partial circular paths.
  • Understanding how to perform calculations with arcs and sectors thus translates to the ability to solve a wide variety of real-world problems.

Recognising and Solving Problems Involving Arcs and Sectors

  • Problems involving arcs and sectors often involve finding the area of sector or length of an arc. The language used in these problems can vary, but look for key terms like “part of a circle”, “sector”, “arc”, or “central angle”.
  • Arc and sector problems can often involve multiple steps and may be combined with other geometric principles, so ensuring a strong foundational knowledge of circles, angles, and other geometric concepts is crucial.

Checking Your Answers

  • Similar to other mathematical problems, always carry out an estimate to check the sensibility of your final answer.
  • Ensure you have used the correct units, particularly when converting between degrees and radians.
  • Rework your calculations when there’s time, to ensure no errors were made in using the arc and sector formulas.