Lesson 12

Radian Sense

  • Let’s get a sense for the sizes of angles measured in radians.

Problem 1

Match each diagram showing a sector with the measure of its central angle in radians.

Circles A through F with sectors shown.

Problem 2

In the circle, sketch a central angle that measures \(\frac{2\pi}{3}\) radians.

Circle with center point.

Problem 3

Angle \(AOC\) has a measure of \(\frac{5\pi}{6}\) radians. The length of arc \(AB\) is \(2\pi\) units and the radius is 12 units. What is the area of sector \(BOC\)?

Angle A O C. Point B falls on perimeter, between points A and C. A B labeled 2 pi. A O labeled 12. Sector B O C shaded.


Problem 4

Calculate the radian measure of a 30 degree angle. Use any method you like, including sketching in the circle diagram provided. Explain or show your reasoning.

Circle with radius 1 unit.
(From Unit 7, Lesson 11.)

Problem 5

Lin thinks that the central angle in circle A is congruent to the central angle in circle B. Do you agree with Lin? Show or explain your reasoning.

circle A

Circle with sector with central angle. Radius 12. Arc measure 4 pi.

circle B

Circle with radius = 3. Sector with length = pi.
(From Unit 7, Lesson 11.)

Problem 6

circle A

Circle radius of 5 units. A 60 degree sector is noted.

circle B

A circle, radius 9 units. A 90 degree sector is noted.

Select all true statements.


The sector in circle B has a larger area than the sector in circle A.


Not taking into account the sectors, circle A and circle B are similar.


The fraction of the circumference taken up by the arc outlining circle A’s sector is smaller than the fraction of the circumference taken up by the arc in circle B.


The ratio of the area of circle A’s sector to its total area is \(\frac16\).


The ratio of circle A’s area to circle B’s area is \(\frac 59\).

(From Unit 7, Lesson 10.)

Problem 7

Match each arc length and radius with the measure of the central angle that defines the arc.

(From Unit 7, Lesson 9.)

Problem 8

Quadrilateral \(ABCD\) is shown with the given angle measures. Select all true statements.

Quadrilateral A B C D inscribed in a circle. Angle A B C is 125 degrees, angle B C D is 40 degrees.

Angle \(A\) measures 140 degrees.


The measures of angle \(A\) and angle \(D\) must add to 180 degrees.


Angle \(A\) measures 55 degrees.


Angle \(D\) measures 55 degrees.


Angle \(D\) measures 40 degrees.

(From Unit 7, Lesson 4.)