R a d i a n M e a s u r e AREA. initial side terminal side radius of circle is r r r arc length is also r r This angle measures 1 radian Given a circle.

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Presentation transcript:

R a d i a n M e a s u r e AREA

initial side terminal side radius of circle is r r r arc length is also r r This angle measures 1 radian Given a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian. Another way to measure angles is using what is called radians.

Arc length s of a circle is found with the following formula: arc lengthradiusmeasure of angle IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA! Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. 3  = 0.52 arc length to find is in black s = r  = 1.56 m What if we have the measure of the angle in degrees? We can't use the formula until we convert to radians, but how? s = r 

We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians. Let's start with the arc length formula s = r  If we look at one revolution around the circle, the arc length would be the circumference. Recall that circumference of a circle is 2  r 2  r = r  cancel the r's This tells us that the radian measure all the way around is 2 . All the way around in degrees is 360°. 2  =  2  radians = 360°

Convert 30° to radians using a conversion fraction. 30° The fraction can be reduced by 2. This would be a simpler conversion fraction. 180°  radians = 180° Can leave with  or use  button on your calculator for decimal. Convert  /3 radians to degrees using a conversion fraction. = radians  0.52 = 60°

Here are the angles in degrees and their corresponding radians. You should become familiar with both

Area of a Sector of a Circle The formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is:  r Again  must be in RADIANS so if it is in degrees you must convert to radians to use the formula. Find the area of the sector if the radius is 3 feet and  = 50° = radians

A Sense of Angle Sizes See if you can guess the size of these angles first in degrees and then in radians. You will be working so much with these angles, you should know them in both degrees and radians.