CIRCULAR MEASURE

When two radii OA and OB are drawn in a circle, the circle is split into two sectors. The smaller sector OAB is called the minor sector. The remainder is the major sector. If we denote the arc length of a sector by s, then we define the angle θ to be 1 radian, when s = r. Radians: Clearly, if θ = 2 radians, the arc length, s = 2r. if θ = 3 radians, the arc length, s = 3r. i.e. it follows that, arc length, s = r θ θ r r s O A B ( With θ measured in radians ).

It follows that if θ = 2π radians, the arc length, s = 2π r. s = r θ Now, since However, 2π r is the circumference of the circle: So: 2π = 360˚ π = 180˚ π2π2 = 90˚ π3π3 = 60˚ π6π6 = 30˚ π4π4 = 45˚ π 180 = 1˚ 1 = 180˚ π c Generally: Note the small ‘c’ to denote radians, though often the abbreviation rads. is used, and for multiples of π, it is usual to use no symbol.

Area of a sector: The minor sector OAB is a fraction of the whole circle. i.e. The area of the sector, A = θ 360 π r 2 × 2π = 360˚ But: A = 1212 r 2 θ θ r r s O A B θ 2π π r 2 × A = ( So, if θ is measured in radians ):

Example 1: A sector OPQ of a circle, radius 4cm, is shown.Given that the angle θ is 1.5 radians, find the perimeter and area of the minor sector. θ 4 4 O P Q Perimeter, p = arc PQ s = r θ Using: = 8 + 6= 14 The area, is given by: A = 1212 r 2 θ A = (1.5) = 8 (1.5) i.e. The perimeter = 14 cm = 12 i.e. The area is 12 cm 2. p = 8 + 4(1.5)

Example 2: The diagram shows the cross section of a large tent. OAB is a sector of a circle, radius 6m. Given that OC = 8m, and the arc length AB is 2π m, find the size of angle AOB and hence find the exact area of the cross section OABC. O 6 C B A 6 8 s = r θUsing arc length, Area of sector OAB, using A = 1212 r 2 θ = 6π Area of triangle OBC, using A = 1212 a b sin C α β = π3π3 = 1212 (6)(8) sin π6π6 = 12 Total area of cross section = (6π + 12) m 2. ( β = – α π2π2 = 2π = 6α π3π3 α = π6π6 ) ( Let AOB = α, BOC = β )

Summary of key points: This PowerPoint produced by R.Collins ; Updated Sep s = r θ π = 180˚ A = 1212 r 2 θ Arc length is given by: Area of a sector is given by: π2π2 = 90˚ π6π6 = 30˚ π4π4 = 45˚ π3π3 = 60˚ Some key angles: Note that for the above results, θ must be in radians.