A part of the circumference of a circle is called an arc. If the angle subtended by the arc at the centre of the circle is θ then the arc length l is given by
The region between the two radii and the arc is called a sector of the circle. The area of the sector of the circle is
Worked Examples
The shaded area shows a segment of a circle of radius 64 cm.
The length of the chord AB is 100 cm.
Find the angle θ, to 2 d.p.
Show meIf AB = 100 cm then, by symmetry, BC = 50 cm.
| sinθ | = |
| θ | = 51.38° |
Find the area of triangle OAB.
Show meThe area of the triangle OAB is × 642 × sin 2θ = 1997 cm2
(This is using the result area of triangle = absinθ, which is covered in another unit.)
Find the area of the sector of the circle with angle 2θ.
Show meThe sector has area × π × 642 = 3673 cm2
Find the area of the segment shaded in the figure.
Show meThe segment has area 3673 − 1997 = 1676 cm2 .
A wooden door wedge is in the shape of a sector of a circle of radius 10 cm with angle 24° and constant thickness 3 cm.
Find the volume of wood used in making the wedge. (Volume is covered in another unit.)
The area of the top face of the wedge is the area of a sector of radius 10 cm and angle 24°.
| Area | = × π × 102 = = 20.94 cm2 | |
| The volume of the wedge | ||
| = Area × 3 = 20π = 62.83 cm3 | ||
Exercises
