Measuring Anglesinin Radians 12 3 EXT Lesson Presentation

Measuring. Anglesinin. Radians 12 -3 -EXT Lesson Presentation Holt. Mc. Dougal Geometry Holt

12 -3 -EXT Measuring Angles in Radians Objectives Use proportions to convert angle measures from degrees to radians. Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians Vocabulary radian Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians One unit of measurement for angles is degrees, which are based on a fraction of a circle. Another unit is called a radian, which is based on the relationship of the radius and arc length of a central angle in a circle. Four concentric circles are shown, with radius 1, 2, 3, and 4. The measure of each arc is 60°. Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians The relationship between the radius and arc length is π 60° linear, with a slope of 2π 360° = 3 , or about 1. 05. The slope represents the ratio of the arc length to the radius. This ratio is the radian measure of the angle, π so 60° is the same as 3 radians. Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians If a central angle θ in a circle of radius r intercepts an arc of length r, the measure of θ is defined as 1 radian. Since the circumference of a circle of radius r is 2πr, an angle representing one complete rotation measures 2π radians, or 360°. 2π radians = 360° and π radians = 180° π radians 1° = 180° and 1 radian = π radians Use these facts to convert between radians and degrees. Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians Remember! Arc length is the distance along an arc measured in linear units. In a circle of radius r, the length of an arc with a m° central angle measure m is L = 2πr 360° Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians Example 1: Converting Degrees to Radians Convert each measure from degrees to radians. A. 85° 17 85° π radians 180° 36 = 17 π 36 π radians 180° 2 = A. 90° 1 90° Holt Mc. Dougal Geometry π 2

12 -3 -EXT Measuring Angles in Radians Helpful Hint Because the radian measure of an angle is related to arc length, the most commonly used angle measures are usually written as fractional multiples of π. Holt Mc. Dougal Geometry

12 -3 -EXT Measuring Angles in Radians Check It Out! Example 1 Convert each measure from degrees to radians. A. – 36° -1 -36° π radians 180° 5 =- π 5 π radians 180° 2 = B. 270° 3 270° Holt Mc. Dougal Geometry 3π 2

12 -3 -EXT Measuring Angles in Radians Example 2: Converting Radians to Degrees Convert each measure from radians to degrees. A. 1 2π 3 2 π radians 3 60 180° radians = 120° 30 180° radians = 30° Π π 6 B. 1 π radians 6 Holt Mc. Dougal Geometry Π

12 -3 -EXT Measuring Angles in Radians Check It Out! Example 2 Convert each measure from radians to degrees. A. 1 5π 6 5 π radians 6 30 Π 180° radians = 150° 180° radians = -135° B. - 3π 4 1 3 π radians 4 Holt Mc. Dougal Geometry 45 Π