IB DP Maths: AI HL复习笔记3.1.3 Arcs & Sectors
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°
Exam 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),
ii)the perimeter of the remaining pizza.
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°
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,
ii)the area Jamie will paint yellow.
Arcs & Sectors Using Radians
How do I use radians to find the length of an arc?
How do I use radians to find the area of a sector?
Worked Example
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