Circle Sectors and Arc Lengths

Arc Length, Area of Sector and Radian Measure

by Mr Wang Kangcheng

​Learning Objectives

• ​find the arc length of a circle by expressing the arc length as a fraction of the circumference of the circle.

• ​find the area of the sector of a circle by expressing the area of a sector as a fraction of the area of the circle.

• ​find the area of a segment of a circle.

• ​convert angular measure from radians to degrees and vice versa.

• ​use the formula

​major sector

​minor sector

​major segment

minor segment

​major arc

minor arc

Poll

Which of the following is the formula for the circumference of a circle?

• $$\pi r$$  

• $$2\pi r$$  

• $$\pi r^2$$  

• $$2\pi r^2$$  

Poll

Which of the following is the formula for the area of a circle?

• $$\pi r$$  

• $$2\pi r$$  

• $$\pi r^2$$  

• $$2\pi r^2$$  

​Work Example 1

(i) ​Find the length of the minor arc AXB.

​Work Example 1

(ii) ​Find the perimeter of the major sector OAYB.

Fill in the Blanks

Find the perimeter of the shaded region, giving your answer in 3 significant figures.

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The figure below shows the design of a logo in which a sector has been removed from the circle, centre O and radius r cm. The length of the major arc PXQ is 36 cm and angle POQ 50^o, find the value of r.

​Practice Now 4

​Radius = 15 cm and the minor arc AQB = 33 cm.

Show that angle AOB is approximately 126.1°.

​Practice Now 4

Hence, find the area of the major sector OAPB.

Fill in the Blanks

Find the length of the major arc AYB.

Multiple Choice

Which is the length of minor arc ACB ?

• $$\frac{2}{3}\pi r$$  

• $$\frac{1}{3}\pi r$$  

• $$2\pi r$$  

• $$r\theta$$  

Fill in the Blanks

Taking  $\pi=\frac{22}{7}$  , calculate the length of arc AB.

Fill in the Blanks

Given that the length of minor arc ACB is 14 cm, find ∠AOB, correct to 1 decimal place.

Fill in the Blanks

The diagram shows a circle with centre O, radius 10 cm and reflex ∠POQ = 315°.  Find the area of the major sector OPRQ.  Leave your answer correct to 3 significant figures.

Fill in the Blanks

A piece of wire is bent to form a sector of a circle of radius 12 cm.  The angle subtended by the wire at the centre of the circle is 205°.  Find the area of the sector, correct to 3 significant figures.

Fill in the Blanks

Find the area of the shaded region ABCD, correct to 3 significant figures.