Lesson Seed: Radian Measures of Angles and Arcs
Common Core Standards:
F.TF.A.1 Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
MP2:
MP4:
MP6:
MP7:
Reason abstractly and quantitatively.
Model with mathematics.
Attend to precision.
Look for and make use of structure.
Common Core Algebra II, Unit 7
Pre-Assessment:
1. Ask students, in pairs, to write down two things they remember about radians. Have
randomly selected pairs share one of their ideas. Allow for critique and amendment.
2. Continue until all ideas have been suggested and vetted. Ensure that these questions have
been addressed:
 “What is a radian?” (A unit of measure for an angle or arc that is related to the radius of
the circle that captures the arc. A one radian arc is exactly one radius of the circle in
length.)
 “How does a radian measure compare to a degree measure?” (The conversion factor from
degrees to radians π/180 or from radians to degrees is 180/π.)
 “What is the difference between a radian measure of an arc and the actual length of an
arc?”(The radian measure is a relative measure of the number of radii of the circle
associated with the angle as a central angle. The actual length is the radian measure
multiplied by the radius length. Note: on a unit circle, the radius is one unit and the
radian measure and the arc length are the same)
Instructional Note: Check for understanding, use the pre-assessment as a formative assessment.
and use this lesson if needed:
Common Core Geometry Wiki Unit 3: G.C.B.5 Measures of Arcs and Radians
Lesson Activities:
1. Once there is a clear understanding of the relationship between degrees and radians,
distribute the sheet (Arcs and Angles on Unit Circle) on the page below. Explain that the
angles, measured from the positive x-axis are 15 degrees apart. Have students, in small
groups, begin the label the arc lengths on the unit circle in radians only in terms of π.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
2. Ask students to look for patterns in the multiples of π. Ask them to consider how they could
find the measure of an arc and the associated central angle in radians in terms of π without
converting from degrees. Allow groups to share and allow for discussion and refinement.
3. Ask, “What is the literal length of the arc?” (This should lead to a discussion of the need to
know the radius of the circle.)
4. Clarify that the circle given is the unit circle, that the radius of the circle is one and the circle
is centered at the origin. Ask students to state, literally, how long each arc would be.
(Students should come to the conclusion that since the radius is one, the literal length of each
arc is exactly the radian measure.)
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
ARCS AND ANGLES ON A CIRCLE
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative
Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.