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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.

When a central angle theta is measured in radians, the length of the arc it cuts off is s = r*theta and the area of the pie-slice sector it bounds is A = (1/2)*r^2*theta. Both formulas come directly from scaling the full circle by the fraction theta/(2*pi), which is why theta must be in radians, not degrees. Because each formula links three quantities, you can solve for whichever one is unknown by substituting the two you know and isolating it.

What you'll learn

Why it matters: Engineers use these formulas to find the length of curved road or railway sections and the floor area of wedge-shaped rooms, while designers use them to lay out the exact size of a slice of a circular logo or pie chart.

Worked example

Problem. A circle has radius r = 9. A sector is formed by a central angle of theta = pi/6 radians. Find both the arc length and the sector area, leaving answers in terms of pi.

  1. Arc length: s = r*theta = 9*(pi/6) = 9*pi/6 = 3*pi/2.
  2. Sector area: A = (1/2)*r^2*theta = (1/2)*(9^2)*(pi/6) = (1/2)*81*(pi/6) = 81*pi/12.
  3. Simplify the area: 81*pi/12 = 27*pi/4.

Answer: Arc length s = 3*pi/2; Sector area A = 27*pi/4

Practice problems

1. A circle has radius r = 5. Find the arc length cut off by a central angle of theta = 2 radians.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A circle has radius r = 5. Find the arc length cut off by a central angle of theta = 2 radians.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use the arc length formula s = r*theta.
  4. Substitute: s = 5*2.
  5. s = 10.
  6. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

2. A sector has radius r = 3 and central angle theta = 4 radians. Find its area.

Show solution
  1. Warm-up: First identify exactly what the question is asking: A sector has radius r = 3 and central angle theta = 4 radians. Find its area.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use the sector area formula A = (1/2)*r^2*theta.
  4. Substitute: A = (1/2)*(3^2)*4 = (1/2)*9*4.
  5. A = 18.
  6. Check the result by substituting or estimating: the response should match 18 and make sense in the original problem.

Answer: 18

3. An arc of length s = 12 is subtended by a central angle theta = 3 radians. Find the radius r.

Show solution
  1. Warm-up: First identify exactly what the question is asking: An arc of length s = 12 is subtended by a central angle theta = 3 radians. Find the radius r.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Start from s = r*theta and solve for r = s/theta.
  4. Substitute: r = 12/3.
  5. r = 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. A circle has radius r = 6. Find the arc length cut off by a central angle of theta = pi/3 radians. Leave your answer in terms of pi.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A circle has radius r = 6. Find the arc length cut off by a central angle of theta = pi/3 radians. Leave your answer in terms of pi.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use s = r*theta.
  4. Substitute: s = 6*(pi/3) = 6*pi/3.
  5. s = 2*pi.
  6. Check the result by substituting or estimating: the response should match 2*pi and make sense in the original problem.

Answer: 2*pi

5. A sector has radius r = 4 and central angle theta = pi/2 radians. Find its area in terms of pi.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A sector has radius r = 4 and central angle theta = pi/2 radians. Find its area in terms of pi.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use A = (1/2)*r^2*theta.
  4. Substitute: A = (1/2)*(4^2)*(pi/2) = (1/2)*16*(pi/2).
  5. A = 4*pi.
  6. Check the result by substituting or estimating: the response should match 4*pi and make sense in the original problem.

Answer: 4*pi

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