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Arc Length and Sector Area

A free Trigonometry lesson from the “Angles, Degrees, and Radians” unit, with a worked example and practice problems including step-by-step solutions.

Angle fluency means moving between degrees, radians, coterminal angles, reference angles, and quadrant signs. These skills turn the unit circle from a memorized chart into a predictable coordinate model.

What you'll learn

Why it matters: Robotics, animation, wheels, gears, and circular motion use angle units, full rotations, and radian formulas.

Worked example

Problem. Arc Length and Sector Area: Find the reference angle for 315 degrees.

  1. Locate the quadrant.
  2. Measure the acute angle to the x-axis.
  3. The reference angle is 45 degrees.

Answer: 45

Practice problems

1. Arc Length and Sector Area: Convert 225 degrees to radians.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Arc Length and Sector Area: Convert 225 degrees to radians.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use 180 degrees = pi radians.
  4. Compute 225 * pi/180.
  5. The result is 5pi/4.
  6. Check the result by substituting or estimating: the response should match 5pi/4 and make sense in the original problem.

Answer: 5pi/4

2. Arc Length and Sector Area: Convert pi/3 radians to degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Arc Length and Sector Area: Convert pi/3 radians to degrees.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use pi radians = 180 degrees.
  4. Match the common unit-circle angle.
  5. Write the degree measure.
  6. Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.

Answer: 60

3. Arc Length and Sector Area: Find the angle from 0 to 360 degrees coterminal with -510 degrees.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Arc Length and Sector Area: Find the angle from 0 to 360 degrees coterminal with -510 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Add or subtract full turns of 360 degrees.
  4. The standard coterminal angle is 210.
  5. The terminal side is unchanged.
  6. Check the result by substituting or estimating: the response should match 210 and make sense in the original problem.

Answer: 210

4. Arc Length and Sector Area: Find the reference angle for 315 degrees.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Arc Length and Sector Area: Find the reference angle for 315 degrees.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Locate the quadrant.
  4. Measure the acute angle to the x-axis.
  5. The reference angle is 45 degrees.
  6. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

5. Arc Length and Sector Area: In Quadrant II, which basic trig function is positive?

Choices: sine · cosine · tangent · cosine and tangent

Show solution
  1. Core Practice: First identify exactly what the question is asking: Arc Length and Sector Area: In Quadrant II, which basic trig function is positive?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Quadrant II has positive y and negative x.
  4. Sine follows y.
  5. Cosine and tangent are negative.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sine

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