IBMathsDPApplications & Interpretation (AI)HLRevision Notes3. Geometry & TrigonometryGeometry ToolkitArcs & Sectors

Arcs & Sectors (DP IB Applications & Interpretation (AI)) : Revision Note

Amber

Author

Amber

Last updated

DP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

Did this video help you?

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 Tips and Tricks

  • 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

Did this video help you?

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

Did this video help you?

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
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Test yourselfFlashcards

Did this page help you?

Previous:Radian MeasureNext:3D Coordinate Geometry
Amber

Author: Amber

Expertise: Maths Content Creator

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

Download notes on Arcs & Sectors