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Radian Measure

On this web page you will investigate using radians to measure angles. This exploration will reinforce the concepts in Lesson 10.2 of Discovering Advanced Algebra: An Investigative Approach.

Sketch

This sketch gives a dynamic representation of the definition of radian: a central angle with measure equal to one radian intercepts an arc that has length equal to the radius of the circle.

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

Investigate

  1. Press Rotate Radius to see the radius of the circle rotate through an angle of 270?? .
  2. Press Wrap Arc to see the radius of the circle wrap around an arc of the circle that intercepts an angle of one radian.

Sketch

This sketch allows you to mark off arcs with measure equal to one radian on a circle. Press the red X to hide the arcs and start over.

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

Investigate

  1. Press 1 Radian to see an arc of measure equal to one radian marked off. About how many degrees are in this arc?
  2. Press Semicircle to mark off a semicircle with one end at point B. About how many radians are there in a semicircle?
  3. Press 1 Circle to mark off a circle based at point B. About how many radians are there in a circle?
  4. Press Go to mark off radians around the circle clockwise. Press the button again to stop. About how many radians are there in two full rotations?

Sketch

This sketch shows a central angle along with its measure in degrees and radians.

Please install Java (version 1.4 or later) to use JavaSketchpad applets.

Investigate

  1. Drag point C so that the angle ABC measures 50??. What is the measure in radians?
  2. Drag point C so that angle ABC measures 3.5 radians. What is the measure in degrees?
  3. Now hide the degree and angle measures. See if you can move point C so that angle ABC has each measure below. Press the Show buttons to see how close your estimate is.

    a. 80??

    b. 0.5 radians

    c. 1 radian

    d. 1.5 radians

    e. 2 radians

  4. Write a formula that gives the measure of an angle in radians if you know its measure in degrees.
  5. Write a formula that gives the measure of an angle in degrees if you know its measure in radians.