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Wheels Distance and Speed Project worksheet for A of F
In Geometry and most everyday applications, angles are measured in degrees. However, radian
measure is another way to measure angles. Using radian measure allows us to write trigonometric
functions as functions not only of angles but also of real numbers in general.
𝜃𝑑 − 𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
𝜃𝑟 − 𝑎𝑛𝑔𝑙𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
360° = 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 (𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 6.28 𝑟𝑎𝑑𝑖𝑎𝑛𝑠)
Note : 6.28 is a real number
-To convert from radians to degrees, multiply the radian measure by
degrees to radians, multiply the degree measure by
𝜋
180°
180°
.
𝜋
Similarly, to convert from
-To find radian measure, use the formula
𝜃𝑟=
𝑠
𝑟
(The formula is valid only if s (arc length) and r (radius) are expressed in the same units)
- You can use the formula for radian measure to find the arc length of a circle.
𝑠 = 𝑟 𝑟
-If a point P moves about the circumference of a circle at a constant speed than linear speed v is given
by
𝑣=
𝑠
(s is arc length and t is the time)
𝑡
-If a point P moves about the circumference of a circle at a constant speed, then the central angle  that
is formed with the terminal side passing through point P also changes over some time t at a constant
speed. The angular speed  (omega) is given by
𝜔=
𝜃𝑟
𝑡
In Exercise 1-5 convert from degrees to radians
1. 30°
2. 285°
3. 185°
4. - 300°
5. - 150°
9. 4.27
10. −8√5
In Exercise 6-10 convert from radians to degrees.
6.
5

7. 36
8
8. −
5
6
In Exercise 11-14 find the measure (in radians) of a central angle  that intercepts an arc of length s on a
circle with radius r. (pay attention to the units)
1
2
12. r =
13. r = 2 m , s = 110 cm
14. r = 24 in. , s = 3 ft.
Angular speed and Linear speed are related through the radius.
If we use the 3 formulas
𝑠 = 𝑟 𝑟 , 𝑣 =
𝑠
𝑡
Explain how 𝑣 = 𝑟𝜔.
, 𝑎𝑛𝑑 𝜔 =
𝜃𝑟
𝑡
in. , s =
3
8
11. r = 5 cm , s = 2 cm
in.
Define Linear Speed-___________________________________________________________________
____________________________________________________________________________________
Define Angular Speed-__________________________________________________________________
____________________________________________________________________________________
15. Find the linear speed of a point that moves with constant speed in a circular motion if the
point travels along the circle an arc length s = 1200 inches in time t = 12 seconds. Round
your answer to two decimal places.
a. Find linear speed in inches per sec
b. Convert linear speed to miles per hour
16. A bicyclist is traveling at an angular speed of
radians per second. How fast is she
traveling in miles per hour if her tires are 27 inches in diameter? Round your answer to one
decimal place.
Collecting Data –
Wheel
Radius
Speed of
Car
Graph the relationship between the wheel radius and the speed of the car
(Hint: put the radius on the x axis)