IBMaths: AI HLDPRevision Notes3. Geometry & Trigonometry3.1 Geometry Toolkit3.1.3 Arcs & Sectors

Arcs & Sectors (DP IB Maths: AI HL)

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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 equals theta over 360 cross times space 2 pi italic space r

    • Where theta is the angle measured in degrees
    • r is the radius
    • This is in the formula booklet, you do not need to remember it

Examiner Tip

  • Make sure that you read the question carefully to determine if you need to calculate the arc length of a sector, the perimeter or something else that incorporates the arc length!

Worked example

A circular pizza has had a slice cut from it, the angle of the slice that was cut was 38 °. The radius of the pizza is 12 cm. Find

 

i)
the length of the outside crust of the slice of pizza (the minor arc),ai-sl-3-1-2-length-arc-we-i

 

ii)
the perimeter of the remaining pizza.ai-sl-3-1-2-length-arcwe-ii

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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 equals theta over 360 cross times pi r squared

    • Where theta is the angle measured in degrees
    • r is the radius
    • This is in the formula booklet, you do not need to remember it

Worked example

Jamie has divided a circle of radius 50 cm into two sectors; a minor sector of angle 100° and a major sector of angle 260°. He is going to paint the minor sector blue and the major sector yellow. Find

i)
the area Jamie will paint blue,

ai-sl-3-1-2-area-sectorwe-i

 

ii)
the area Jamie will paint yellow. ai-sl-3-1-2-area-sectorweii

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Arcs & Sectors Using Radians

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

  • As the radian measure for a full turn is 2 straight pi, the fraction of the circle becomes fraction numerator theta over denominator 2 pi end fraction
  • Working in radians, the formula for the length of an arc will become

l equals fraction numerator theta over denominator 2 pi end fraction blank cross times 2 pi space r

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

l space equals space r theta space

    • theta is the angle measured in radians
    • r is the radius
    • This is given in the formula booklet, you do not need to remember it

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

  • As the radian measure for a full turn is 2 straight pi, the fraction of the circle becomes fraction numerator theta over denominator 2 pi end fraction
  • Working in radians, the formula for the area of a sector will become

A equals fraction numerator theta over denominator 2 pi end fraction blank cross times pi space r squared

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

A equals 1 half space r squared space theta

    • theta is the angle measured in radians
    • r is the radius
    • This is given in the formula booklet, you do not need to remember it

Worked example

A slice of cake forms a sector of a circle with an angle of straight pi over 6 radians and radius of 7 cm. Find the area of the surface of the slice of cake and its perimeter.

 aa-sl-3-1-3-arcs-and-sectors-using-radians-we-solution

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