St. James 03-18-2025 Circles Radian Length & Degree Conversion

● I can define the parts of a circle
● I can define and measure in a circle
● I can find the measure of segments of a circle
● I can discover the properties of tangent lines
● I can find the measure of angles inside and outside of a circle
● I can find missing measures of angles and arcs using circle theorems.

● I can explain radian angle measures in terms of a circular radius and
arc length

Objectives

• Efficiency: Radians simplify circular motion calculations.

• Real-World Use: Engineering, physics, navigation, and robotics rely on radians for precision.

Key Idea: Radians measure angles based on arc length. They are essential in fields like robotics, astronomy, and architecture.

Why use Radians?!

Converting 200 degrees to radians

​Converting Radians to Degrees
2

​r/ = 200/180

r = 200/180

r = 10/9
EXACT Answer simplified

​1/ 2 = θ/180
θ = 1/2 (180)
θ = 90

Match

Match the following

90 Degrees

-270 Degrees

250 Degrees

-110 Degrees

270 Degrees

• 90 Degrees

• -270 Degrees

• 250 Degrees

• -110 Degrees

• 270 Degrees

Converting 200 degrees to radians

​Converting Radians to Degrees
2

​r/ = 200/180

r = 200/180

r = 10/9
EXACT Answer simplified

​1/ 2 = θ/180
θ = 1/2 (180)
θ = 90

Open Ended

Convert 36 degrees to radians.

Multiple Choice

$$180\degree$$  = _________radians

• $$\frac{\pi}{2}$$  

• $$\pi$$  

• $$2\pi$$  

• $$\infty$$  

Multiple Choice

What is  $$225\degree$$   in radians? 

• $$\frac{7\pi}{4}$$  

• $$\frac{5\pi}{4}$$  

• $$\frac{3\pi}{4}$$  

• $$\frac{5\pi}{3}$$  

Multiple Choice

What is $$270\degree$$  in radians?

• $$\frac{\pi}{4}$$

• $$\frac{3\pi}{2}$$

• $$\frac{3\pi}{4}$$

• $$\frac{\pi}{2}$$

Multiple Choice

Convert 150⁰ to radians

• $$\frac{5\pi}{6}$$

• $$\frac{3\pi}{4}$$

• $$\frac{7\pi}{6}$$

• $$\frac{2\pi}{3}$$

Multiple Choice

Convert 45 degrees to radians.

• $$\frac{\pi}{4}$$  

• $$\frac{4}{\pi}$$  

• $$\frac{4\pi}{5}$$  

• $$\frac{5}{4\pi}$$  

• Efficiency: Radians simplify circular motion calculations.

• Real-World Use: Engineering, physics, navigation, and robotics rely on radians for precision.

Key Idea: Radians measure angles based on arc length. They are essential in fields like robotics, astronomy, and architecture.

Why use Radians?!

​

Multiple Choice

Find the arc length indicated by the bolded arc.

• 8.3 in

• 7.2 in

• 6.8 in

• 6.4 in

Multiple Choice

Find the arc length indicated by the bolded arc.

• 17.3 km

• 79.1 km

• 95.0 km

• 99.0 km

Multiple Choice

Find the arc length indicated by the bolded arc.

• 8.4 mi

• 9.4 mi

• 60.2 mi

• 117.8 mi

Multiple Choice

Find the arc length.

• 3π/4 ft

• 39π/4 ft

• 13π/4 ft

• 4π/13 ft

Multiple Choice

The diameter of a circle is 8 centimeters. A central angle of the circle intercepts an arc of 12 centimeters. What is the radian measure of the angle?

• 3

• 4

• 8π

• 3/2

Multiple Choice

What is the approximate length of the arc subtended by an angle of 4π/3 radians on a circle with a radius of 6.00 meters?

• 12.57 meters

• 14.14 meters

• 25.13 meters

• 28.27 meters

Multiple Choice

Convert -520^o into Radians

• -17π / 4

• -21π / 6

• -26π / 9

• -23π / 6

Match

Match the following angles to their standard position.

−340°

150°

−185°

300°

−260°

• −340°

• 150°

• −185°

• 300°

• −260°

Match

Match the following angles correctly.

• $$\frac{3\pi}{2}$$

• $$\frac{2\pi}{3}$$

• $$\pi$$

• $$2\pi$$

• $$\frac{3\pi}{4}$$

270 degrees

120 degrees

180 degrees

360 degrees

135 degrees

• 270 degrees

• 120 degrees

• 180 degrees

• 360 degrees

• 135 degrees