Radians and Circles

A brief overveiw of radians and questions that could be found in the C2 maths AS exam along with how to solve them

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What is a radian

What is a radian?

A radian is an angle at the center of a circle where the arc of the circle is equal to the length of the radius of the circle. 

This is one radian. the radius is equal to the arc length meaning radians differ depending on the size of the circle unlike degrees

(http://intmstat.com/trigonometric-functions/Image1961.gif)

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Converting Degrees and Radians

Changing Degrees to Radians

Say you have a circle with a 60 degree angle. To convert this to radians you would multiply 60 by (pi / 180)

60 degrees = 60 x( pi / 180 ) 

                  = 60pi / 180

                  = pi / 3

To reverse Radians to Degrees you simply multiply the radian by 180 / pi 

2 pi / 5 = (2pi /5) x (180 / pi)

           = 72 degree 

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Sectors of Circles

Sector of a Circle

The equation for working out the area of a sector of a circle is:

Area = 1/2 x r^2 x theta (converting the angle into radians)

if r = 8 then the area of sector OQP = 1/2 x 8^2 x (45 x pi/180)

                                                     = 1/2 x 64 x pi/4

                                                     = 25.13cm^2 or 8pi (remember units)

         (http://image.tutorvista.com/Qimages/QD/51094.gif)

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Segments of Circles

Segment of a Circle

The equation for working out the area of a segment of a circle is:

Area = 1/2 x r^2 x (theta - Sin theta) 

if r = 4 and theta = 40 degrees then the area of the segment =

1/2 x 16 x (40 x pi/180 - Sin (40 x pi/180))

= 8 (0.7 - 0.01 )

= 5.52 cm^2

(http://www.tutornext.com/system/files/u37/Area%20of%20Segment%20of%20a%20circle.JPG)

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Exam question practice

Exam Question 

A sector of a circle of radius 6cm has a radius of 6cm has an angle of 1.6 radians.

Find the area of the sector 

Hence find the area of the segment                                                                    [5]

Area of the sector = 1/2 x 6^2 x 1.6

                             = 28.8 cm^2

Area of the segment = 1/2 x 6^2 x (1.6 - Sine (1.6 x 180/pi))

                                 = 10.8 cm^2


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Comments

Inmyimpala

I actually understand how to convert to radians now. Thank you :)

Inmyimpala

I actually understand how to convert to radians now. Thank you :)

Olivia

Thanks this was very helpful :)

daviesg

A useful set of cards on circle measures

Crimsix

you see for slide 4, does you calculator have to be set to radians?

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