Lesson 11

In this activity, students find arc lengths for common angle measurements in a circle with a radius of 1 unit. This will be helpful when arc lengths in unit circles are used as radian angle measurements in an upcoming activity.

Ask students to share their strategies, especially for the 225 degree angle. Some students may create the fraction \(\frac{225}{360}\) and multiply this fraction by \(2\pi\), writing it in equivalent form with smaller numbers either before or after multiplying. Others may observe that 225 is the sum of 180 and 45, so they can add \(\pi\) and \(\frac{\pi}{4}\) to get the answer.

In this activity, students examine and complete a narrative proving that the length of the arc intercepted by a central angle is proportional to the radius of the circle. This will lead directly to the definition of radian measure in the next activity. Analyzing ratios that are invariant under dilation and giving them names is analogous to defining the trigonometric ratios of similar right triangles in a previous unit.

If students are stuck trying to show that \(\frac{\ell}{r}=\frac{L}{R}\), ask if they can rewrite \(\frac{L}{R}\) by substituting in other expressions for \(L\) and \(R\).

The goal is to ensure that students recognize that for a given angle, the length of the arc is proportional to the radius. Ask students, “What does it mean for 2 quantities to be proportional?” (It means that there is some constant multiplier between them.)

Then, display this image for all to see.

Ask:

• “What is the multiplier, or constant of proportionality, between the radius and arc length for these 90 degree central angles? That is, we’re looking for a value \(w\) such that \(\ell=wr\) and \(L=wR\).” (The value of \(w\) is \(\frac{\pi}{2}\).)
• “How does this example relate to what you proved in the activity?” (We proved that the ratio \(\frac{\ell}{r}\) is equal to the value \(\frac{L}{R}\). In this example, we showed both are equal to \(\frac{\pi}{2}\).)

In this activity, students learn the definition of the radian measure of an angle. They observe relationships between the radius of a circle and the length of an arc on that circle, and they determine that 180 degrees is equivalent to \(\pi\) radians.

Monitor for a variety of strategies for the last problem, including calculations based on the definition of radian measure and proportional reasoning using the equivalence of 180 degrees and \(\pi\) radians.

Ask students to list some different units for measuring lengths (miles, kilometers, light years) and volumes (gallons, liters, cubic feet). Tell students that angles also have different units of measurement, and that they will learn about a new angle measurement unit in this activity.

Remind students that they proved that for central angles in circles, arc length and radius are proportional. That is, for a given angle, the ratio of the arc length defined by the angle to the circle’s radius is a constant value. This is such a useful property that it is used as a way to measure angles.

Give students a few minutes to work, then pull the class back together to ensure all students have calculated the measure of the angle in the first part as 1 radian.

If students are confused by the fact that their radian measures of angles are the same as the arc lengths they’ve calculated, remind them that the radius of the circle is 1 unit, so it makes sense that when we calculate the ratio of arc length to radius for angles in these circles, the answer is the arc length divided by 1—that is, it’s simply the arc length.

The goal is to discuss the idea that by relating arc length to a circle’s radius, the radian measure of an angle shows how many radii make up the arc defined by a central angle.

Ask students to share their strategies for the last problem. If possible, select one student who calculated that the circumference of the circle in the activity is \(2\pi\) units, and another who reasoned that 360 degrees is 2 times 180 degrees, so the radian measure of a 360 degree angle must be twice that of a 180 degree angle. Then ask students:

• “Does the word radian seem to relate to a particular part of a circle?” (The word radian sounds like radius.)
• “How is a radian connected to the radius?” (The radian measure of an angle is the ratio of the arc length to the radius, so it tells how many radii fit along the arc defined by the angle.)
• “How did your answers to the second and third questions connect?” (The string measurement showed that the arc length was about 3 units in length. The radian measure of the angle ended up being \(\pi\) or 3.14 units, which matches our approximate string measurements.)
• “What if the circle had a radius of 2 units instead of 1? How would the radian measures change?” (They wouldn’t change. The arc length to radius ratio is constant no matter the size of the circle.)
• “We defined the radian measure of an angle as \(\theta = \frac{\text{arc length}}{\text{radius}}\). How does this relate to the idea that the arc length and radius are proportional?” (We can think of the angle measure as being the constant of proportionality.)