Arcs & Sectors (Cambridge International IGCSE Further Maths)

Length of an Arc

What is an arc?

• An arc is a part of the circumference of a circle

  • It is easiest to think of it as the crust of a single slice of pizza

• The length of an arc depends of the size of the angle at the centre of the circle

• If the angle at the centre is less than 180° then the arc is known as a minor arc

  • This could be considered as the crust of a single slice of pizza

• If the angle at the centre is more than 180° then the arc is known as a major arc

  • This could be considered as the crust of the remaining pizza after a slice has been taken away

How do I find the length of an arc?

• The length of an arc is simply a fraction of the circumference of a circle

  • The fraction can be found by dividing the angle at the centre by 360°

• The formula for the length, $l$, of an arc is

$l=\frac{θ}{360}\times2πr$

• Where $θ$ is the angle measured in degrees

  • $r$ is the radius

How do I use radians to find the length of an arc?

• As the radian measure for a full turn is $2π$, the fraction of the circle becomes $\frac{θ}{2π}$
• Working in radians, the formula for the length of an arc will become

$l=\frac{θ}{2π}\times2πr$

• Simplifying, the formula for the length, $l$, of an arc is

$l=rθ$

• • $θ$ is the angle measured in radians
  • $r$ is the radius

Worked Example A circular pizza has had a slice cut from it, the angle of the slice that was cut was $\frac{π}{6}$ rad.

The radius of the pizza is 12 cm. Find

i) the length of the outside crust of the slice of pizza (the minor arc),

A simple diagram will help

• The formula for the length of an arc, where the angle is in radians is *$s=rθ$

$s=12\times\frac{π}{6}$

Final answer: $2π\mathrm{cm}$

ii) the perimeter of the remaining pizza.

A diagram will help consider where the perimeter is

Find the angle for the major arc, by subtracting from the angle in a full circle

$2π-\frac{π}{6}=\frac{11π}{6}$

• Use the formula for the length of an arc, $s=rθ$, to find the curved length of the perimeter, the major arc* $M$

$M=12\times\frac{11π}{6}=22π$

• As we are finding the perimeter of the whole shape, we need to add on the two straight lengths formed by the slice which has been cut out *

$22π+12+12$

$22π+24\mathrm{cm}$ Unless asked to otherwise, it is best to give answers in an exact form

Area of a Sector

What is a sector?

• A sector is a part of a circle enclosed by two radii (radiuses) and an arc

  • It is easier to think of this as the shape of a single slice of pizza

• The area of a sector depends of the size of the angle at the centre of the sector

• If the angle at the centre is less than 180° then the sector is known as a minor sector

  • This could be considered as the shape of a single slice of pizza

• If the angle at the centre is more than 180° then the sector is known as a major sector

  • This could be considered as the shape of the remaining pizza after a slice has been taken away

How do I find the area of a sector?

• The area of a sector is simply a fraction of the area of the whole circle

  • The fraction can be found by dividing the angle at the centre by 360°

• The formula for the area, $A$, of a sector is

$A=\frac{θ}{360}\timesπr^{2}$

• Where $θ$ is the angle measured in degrees

  • $r$ is the radius

How do I use radians to find the area of a sector?

• As the radian measure for a full turn (360°) is $2π$, the fraction of the circle becomes $\frac{θ}{2π}$
• Working in radians, the formula for the area of a sector will become

$A=\frac{θ}{2π}\timesπr^{2}$

• Simplifying, the formula for the area, $A$, of a sector is

$A=\frac{1}{2}r^{2}θ$

• $θ$ is the angle measured in radians

  • $r$ is the radius

Exam Hint

• These formulae are not given to you - you need to remember them!

• Make sure that you read the question carefully to determine exactly what you need to calculate

  • the arc length or area of a sector,
  • the perimeter or area of a compound shape,
  • or something else that incorporates the arc length or area!

Worked Example A sector of radius 6 cm has an area of 30 cm². Find the angle at the centre of the sector in radians. Draw a diagram to help

Use the formula for the area of a sector, where the angle is in radians; $A=\frac{1}{2}r^{2}θ$

$30=\frac{1}{2}\times6^{2}\timesθ$

• Solve for $θ$ *

$θ=\frac{30\times2}{6^{2}}$

$\frac{5}{3}\mathrm{radians}$