You have taken your little cousin to the amusement park for the day. While there, she decides she would like a ride on the carousel. After the ride, she excitedly bounces over to you. She is amazed that she went around, but in a way "didn't go anywhere", since she ended up where she started.
"How far did I go when I was halfway around the turn?", she asks.
You know that the radius of the carousel is 7 meters. Can you tell your little cousin how far she went in one half of a turn around the ride?
At the end of this Concept, you'll be able to do just that.
James Sousa Example: Arc Length and Application of Arc Length
This results in a formula that can be used to calculate the length of any arc.
The free-throw line on an NCAA basketball court is 12 ft wide. In international competition, it is only about 11.81 ft. How much longer is the half circle above the free-throw line on the NCAA court?
Solution: Find both arc lengths.
This is approximately 0.3 ft, or about 3.6 inches longer.
Two connected gears are rotating. The smaller gear has a radius of 4 inches and the larger gear’s radius is 7 inches. What is the angle through which the larger gear has rotated when the smaller gear has made one complete rotation?
Solution: Because the blue gear performs one complete rotation, the length of the arc traveled is:
So the angle is approximately 3.6 radians.
The radius of a standard car tire is 27.94 cm. How far does a car go in one revolution of the tire?
Solution: Since the distance traveled by the tire is equal to the distance around the tire, we can use the circumference of the tire to answer the question.
Arc: An arc is a length segment measured around the circumference of a circle.
2. If an object with a radius of 10 cm spins so that its arc covers 54 cm, what is the change in angle of the object?
The disk moves 5.4 radians, which is a little less than a complete rotation, since a complete rotation is approximately 6.28 radians.
A point on the disk turns 7.065 inches.
Concept Problem Solution
The radius of a carousel is 8 meters. Use this information to answer questions 1-3.
- You are half way around the carousel. How far did you travel?
- You are all the way around the carousel. How far did you travel?
- You have now traveled all the way around the carousel twice. How far did you travel?
A pizza has a radius of 10in. Use this information to answer questions 4-6.
- A slice is removed. The length of the crust of the missing slice is 3in. What is the central angle of the missing slice?
- You eat three pieces with a central angle of 4π5. What is the length of the crust you ate?
- A large pizza has a radius of 12in. What is the length of the crust of half of the large pizza?
The diameter of a tire is 35in. Use this information to answer questions 7-10.
- What is the length around the whole tire?
- The tire travels one mile (5280 ft). How many revolutions did the tire make?
- You roll the tire so it rotates 7π radians. How far did it move?
- The tire travels half a mile. How many radians did the tire rotate?
Consider a standard 12 hour clock like the one below with a radius of 5 inches. Use this to answer questions 11-15.
- What is the length of the arc between the 3 and the 7?
- What is the length of the arc between the 3 and the 2?
- It is 12:30. What is the length of the arc between the minute and hour hands?
- It is 7:20. What is the length of the arc between the minute and hour hands?
- It is 1:25. What is the length of the arc between the minute and hour hands?