CIRCLES, SECTORS AND RADIANS SECTORS The non-shaded area of the circle shown below is called a SECTOR. 90 0 R0cm In this example the sector subtends a right-angle (90 0 ) at the centre of the circle. The non-shaded area would still be a sector if the angle at the centre of the circle was larger, or smaller, than a right-angle (90 0 ). We can see that the non-shaded sector is a quarter of the circle, so its area is one quarter of the total area of the circle. Area of a sector = ( π R ) for this example = π 0 = 00π = 5π cm Since, in this example, the angle subtended by the sector at the centre of the circle is 90 0 and the angle for a full circle 0 we can calculate the area of the sector as follows. Area of sector = 90 ( πr ) 90 = π0 = 00π = 5π cm same as before. The same argument applies for angles other than 90 0 and we can state a general formula as: Area of sector = ( πr ) Where is the angle (in degrees) subtended by the sector at the centre of the circle. Page of 0
Exercise Complete the following table: Radius Area of sector a) 0cm 60 0 b) 5mm 00 0 c) 0mm 50π mm d) 30 0 75π mm Now check your answers So far we have measured the angle, subtended by the sector, in degrees. RADIANS Another unit of angular measure, used frequently in engineering, is the RADIAN. We are now going to discover how we can calculate the area of a sector when the angle it subtends is measured in radians. Let s remind ourselves what a radian is. A radian is defined as: The angle ( ) subtended at the centre of a circle by an arc of the circle equal in length to the radius. Now, how many radians are there in a complete circle you may ask yourself? Well, the circumference of a circle is π times the radius that is π R, and the angle subtended by one radian is equal to one radius πr R. So the number of radians in a complete circle is = π radians, or to put it another way, π R radians = 0 Page of 0
Exercise Complete the table a) π radians 0 b) π radians 0 c) radians 90 0 d) radians 5 0 e) radian.. 0 Now check your answers. Area of the non-shaded sector is: 90 0 R0cm 90 Area = x (π 0 ) = 00π = 5π cm But we have previously discovered that 90 0 = π radians So we can also say: And 0 = π radians π Area = radians ( π0 ) πradians = 00 5 π = cm the same as before. Page 3 of 0
So it would seem reasonable to assume that: Area = radians πr πradians = π x R π = R when is in radians. Area of sector = R when is in radians. Exercise 3 Complete the following table: Angle Radius Area of sector A 0.8 rads 0mm..mm B..rads 0mm 50π mm C π..mm 00π mm rads Now check your answers. Exercise Calculate the shaded area of the optical shutter blade and convert given are radians ( c ). Dimensions in millimetres. to degrees of arc. The angles Now check your answers. Page of 0
ANSWERS Exercise Radius Area of sector a) 0cm 60 0 6.67π cm 5.37π cm b) 5mm 00 0 37.π mm 09mm c) 0mm 80 0 50π mm d) 30mm 30 0 75π mm The Answers are in bold. a) Area of a sector = x (π R ) 60 = Xπ 0 = 6 x 00π =6.67π cm b) Area of a sector = x (π R ) = 00 π 5 00 = 65π = 37.π mm c) Area of sector = x (π R ) 50π = X(π 0 ) 50 π 00π = = 80 0 Page 5 of 0
d) Area of sector = x (π R ) 75π = 30 πr 75π π 30 = R R = 900 Now return to the text. R = 30mm Exercise a) π radians 0 b) π radians 80 0 c) π 90 0 radians d) π radians 5 0 e) radian 57.3. 0 The Answers are in bold. a) To start you off, you have been given π radians = 0 b) If π radians = 0 Then π radians = = 80 0 c) If 80 0 = π radians Then 90 0 = π radians d) If 80 0 = π radians Then 5 0 = π radians Similarly 60 0 = 3 π radians These are useful to remember 30 0 = 6 π radians Page 6 of 0
e) If π radians = 80 0 0 80 then radian = = 57.3 0 π 57.3 0 is an easy figure to remember and is accurate for most practical purposes. Where greater accuracy is required, use conversion tables or a scientific calculator. Now return to the text. Exercise 3 Angle Radius Area of sector A 0.8 rads 0mm 60mm B.π.rads 0mm 50π mm C π.0.mm 00π mm rads The Answers are in bold. a) Area of a sector = R = 0 0. 8 = 00 0. 8 = 60mm b) Area of a sector = R 50π mm = 0 50π ( 0 ) = 50π = 50 = π radians (or 80 0 ) Page 7 of 0
c) Area of a sector = R 00π mm = R π 00π = R ( π ) 00 X = R 600 = R R = 0 mm Now return to the text. Exercise The shutter blade is made up from a number of sectors with a common centre and it is symmetrical about its centre lines. a) First let s find the overall blank area. Total area = R = 75. 06 = 565. 06 = 98.5mm Page 8 of 0
Area A = R =. 06 =. 06 = 76.3mm Shutter blank area is the difference between these two areas. Shutter blank area = 98.5 76.3 = 90.93 mm b) Now let s find the area of the window. Again, this is the difference of two sectors Area of larger sector = R = 65 0. 7 = 5 0. 7 = 78.75mm Page 9 of 0
Area of smaller sector = R = 5 0. 7 = 05 0. 7 = 708.75mm The window area is the difference between these areas. Window area = 78.75 708.75 = 770mm c) To find the shaded area of the optical shutter, we take the window area from the shutter blank area. Shaded are = 90.93 770 = 3.93mm d) Finally we have to convert to degrees of arc. ( is given as.06 radians) Remember we have found that radian = 57.3 0 To convert radians to degrees, multiply by 57.3 To convert degrees to radians, divide by 57.3 So.06 radians =.06 x 57.3 = 60.7 0 Page 0 of 0