Radians and Arc Length

Geoffrey, Olivia, Melanie, Sara

Arc Lengths

Central Angle

An angle whose vertex is at the center of the circle.

***The measure of the intercepted arc is equal to the measure of the central angle.***

Inscribed Angle

An angle whose vertex lies on the circle.

***The measure of the intercepted arc is double the measure of the inscribed angle.***

Circumference of a Circle : πd = 2πr

Area of a Circle : πr²

*** d = distance, r = radius ***


Arc : part of the circle between two points on the circle.

Arc Length : the length of the arc between two points on a circle

*** Fraction of the circumference ***


••• Formula : (measure of arc/360º) x 2πr •••

Find the values of the different arcs - Keep in the values of π

***** Arc Length Formula : (measure of arc/360º) x 2πr *****


Arc AB

  • The angle is 105 so the distance across it is 105.
  • Then divide (105÷360) which simplifies to (7/24).
  • You then multiply. (7/24)(2) and add pi on the end of the number.
  • The final answer : 7/12π


Arc BC

  • The angle is 50º so the distance across it is 50º.
  • Then divide (50÷360) which simplifies to (5/36).
  • You then multiply (5/36)(2) and add pi on the end of the number.
  • The final answer : 5/18π


Arc ED

  • The angle is 120º so the distance across it is 120º.
  • Then divide (120÷360) which simplifies to (1/3).
  • You then multiply (1/3)(2) and add pi on the end of the number.
  • The final answer : 2/3π

Length of an arc that subtends a central angle | Circles | Geometry | Khan Academy

Radians

What is radian?

Radian is a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius.

Converting between degrees and radians

*** Remember 2π = 360º and that π = 180º ***


To convert radians to degrees multiply your given amount of radians by (180/π)


To convert degrees to radians multiply your given amount of degrees by (π/180)

  • When you multiply the π can cancel out.
  • Radian is usually represented by ∂ or π



Change degree to radian

45º

  • (45 degrees) • (π radian/180 degrees)
  • When you multiply the 2 degrees will cancel out since one is on the top and the other is on the bottom and are left with just radian.
  • You end up with 45π/180 radian which can simplify to 4π radian


Change radian to degree

π/3 radians

  • (π/3 radians) • (180 degrees/π radian)
  • When you multiply the 2 radians will cancel out since one is on the top and the other is on the bottom and are left with just degrees.
  • You end up with 180π/3 degrees which can simplify to 60πº

Introduction to radians | Unit circle definition of trig functions | Trigonometry | Khan Academy
Example: Converting degrees to radians | Trigonometry | Khan Academy
Radian and degree | Unit circle definition of trig functions | Trigonometry | Khan Academy