A New Way to Measure Angles

ID: buhol-hilig
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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSG-C.B.5HSA-SSE.A.1.bHSG-C.A.1
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19 questions

A New Way to Measure Angles

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A One-Unit Radius

A circle has radius 1 unit. Find the length of the arc defined by each of these central angles. Give your answers in terms of π\pi.

1) 180 degrees

2) 45 degrees

3) 270 degrees

4) 225 degreees

5) 360 degrees

A Constant Ratio

Diego and Lin are looking at 2 circles.

Diego says, “It seems like for a given central angle, the arc length is proportional to the radius. That is, the ratio r\frac{\ell}{r} has the same value as the ratio LR\frac{L}{R} because they have the same central angle measure. Can we prove that this is true?”

Lin says, “The big circle is a dilation of the small circle. If kk is the scale factor, then R=krR=kr.”

Diego says, “The arc length in the small circle is =θ3602πr\ell = \frac{\theta}{360}\cdot 2\pi r. In the large circle, it’s L=θ3602πRL = \frac{\theta}{360} \cdot 2\pi R. We can rewrite that as L=θ3602πrkL = \frac{\theta}{360} \cdot 2\pi rk. So L=kL = k\ell.”

Lin says, “Okay, from here I can show that r\frac{\ell}{r} and LR\frac{L}{R} are equivalent.”

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6) How does Lin know that the big circle is a dilation of the small circle?

7) How does Lin know that R=krR = kr?

8) Why could Diego write =θ3602πr\ell = \frac{\theta}{360} \cdot 2\pi r?

9) When Diego says that L=kL = k\ell, what does that mean in words?

10) Why could Diego say that L=kL = k\ell?

11) How can Lin show that r=LR\frac{\ell}{r} = \frac{L}{R}?

Defining Radians

Suppose we have a circle that has a central angle. The radian\textbf{radian} measure of the angle is the ratio of the length of the arc defined by the angle to the circle’s radius. That is, θ = arc lengthradius\theta \ = \ \frac{\text{arc length}}{\text{radius}}.

The image shows a circle with radius 1 unit.

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12) Cut a piece of string that is the length of the radius of this circle. Use the string to mark an arc on the circle that is the same length as the radius.

13) Draw the central angle defined by the arc.

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14) Use the definition of radian to calculate the radian measure of the central angle you drew. Round to the nearest whole degree.

15) Draw a 180 degree central angle (a diameter) in the circle. Use your 1-unit piece of string to measure the approximate length of the arc defined by this angle.

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16) What is the approximate length of the arc defined by this angle? Round your answer to the nearest whole number.

17) Calculate the radian measure of the 180 degree angle. Give your answer in terms of π\pi.

18) Round your answer to the nearest hundredth.

19) Calculate the radian measure of a 360-degree angle. Write your answer in terms of π\pi.