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Radian Measure

Grade 11 · Trigonometry · Worksheet 1

  1. Liam is designing a circular running track for his school's new athletic facility. The track has a radius of 45 meters. He needs to install special rubberized surfacing along an arc that spans an angle of 2.1 radians. What length of surfacing material (in meters) should Liam order for this arc section?
    Answer: ______________
  2. 2π/3 radians × (180/π) = ? Answer: ______________
  3. Charlotte is designing a circular fountain for a park. The fountain has a radius of 9 meters, and she wants to place a decorative tile border along an arc that corresponds to a central angle of 5π/7 radians. What is the exact length of the arc (in meters) that will have the tile border? (Express your answer in terms of π.)
    Answer: ______________
  4. A circular sector has a central angle of 2.4 radians and a radius of 8 cm. What is the exact arc length of this sector? Answer: ______________
  5. A circular sector has area 24π cm² and radius 8 cm. Find the central angle in radians: θ = ? Answer: ______________
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Answer Key & Explanations

Radian Measure · Grade 11 · Worksheet 1

  1. Liam is designing a circular running track for his school's new athletic facility. The track has a radius of 45 meters. He needs to install special rubberized surfacing along an arc that spans an angle of 2.1 radians. What length of surfacing material (in meters) should Liam order for this arc section? Answer: 94.5 Solution: To find the length of an arc when the radius and central angle in radians are given, we use the formula: Arc length = radius × angle in radians Identify the known values. The radius of the circular track is 45 meters.
    Full step-by-step solution

    To find the length of an arc when the radius and central angle in radians are given, we use the formula: Arc length = radius × angle in radians Step 1: Identify the known values. The radius of the circular track is 45 meters. The angle of the arc is 2.1 radians. Step 2: Apply the formula. Arc length = 45 × 2.1 Step 3: Perform the multiplication. First, multiply 45 by 2. 45 × 2 = 90 Next, multiply 45 by 0.1. 45 × 0.1 = 4.5 Now, add the two results together. 90 + 4.5 = 94.5 Step 4: State the final answer. Therefore, the length of surfacing material Liam should order is 94.5 meters.

  2. 2π/3 radians × (180/π) = ? Answer: 120 Solution: 2π/3 radians × (180/π) = ? Write the expression clearly. (2π/3) × (180/π) Notice that π appears in both numerator and denominator.
    Full step-by-step solution

    Let's solve step by step. We are given: 2π/3 radians × (180/π) = ? Step 1: Write the expression clearly. (2π/3) × (180/π) Step 2: Notice that π appears in both numerator and denominator. We can cancel π from top and bottom: (2/3) × (180/1) Step 3: Multiply 2/3 by 180. First, 2 × 180 = 360. Then divide by 3: 360 / 3 = 120. Step 4: State the final answer. The result is 120. So, 2π/3 radians equals 120 degrees.

  3. Charlotte is designing a circular fountain for a park. The fountain has a radius of 9 meters, and she wants to place a decorative tile border along an arc that corresponds to a central angle of 5π/7 radians. What is the exact length of the arc (in meters) that will have the tile border? (Express your answer in terms of π.) Answer: (45π/7) meters Solution: Identify the given values. The radius r = 9 meters, and the central angle θ = 5π/7 radians. Recall the arc length formula: s = rθ, where s is the arc length, r is the radius, and θ is the central angle in radians.
    Full step-by-step solution

    Step 1: Identify the given values. The radius r = 9 meters, and the central angle θ = 5π/7 radians. Step 2: Recall the arc length formula: s = rθ, where s is the arc length, r is the radius, and θ is the central angle in radians. Step 3: Substitute the values: s = 9 × (5π/7). Step 4: Multiply: s = (9 × 5π)/7 = 45π/7. Step 5: The exact arc length is 45π/7 meters. The answer is (45π/7) meters.

  4. A circular sector has a central angle of 2.4 radians and a radius of 8 cm. What is the exact arc length of this sector? Answer: 19.2 cm Solution: We have a circular sector with radius r = 8 cm and central angle θ = 2.4 radians. We need the arc length. Recall the formula for arc length.
    Full step-by-step solution

    Step 1: Understand the problem. We have a circular sector with radius r = 8 cm and central angle θ = 2.4 radians. We need the arc length. Step 2: Recall the formula for arc length. For a circle, arc length s is given by: s = r × θ where r is the radius and θ is the central angle in radians. Step 3: Substitute the given values into the formula. r = 8 cm θ = 2.4 radians So: s = 8 × 2.4 Step 4: Perform the multiplication. 8 × 2.4 = 19.2 Step 5: State the final answer with units. The exact arc length is 19.2 cm. This matches the correct answer.

  5. A circular sector has area 24π cm² and radius 8 cm. Find the central angle in radians: θ = ? Answer: 3π/4 Solution: The formula for area of a circular sector is A = (1/2)r²θ Substitute the given values: 24π = (1/2)(8²)θ Simplify: 24π = (1/2)(64)θ Calculate: 24π = 32θ Solve for θ: θ = 24π/32 Simplify the fraction: θ = 3π/4 The central angle is 3π/4 radians.
    Full step-by-step solution

    Step 1: The formula for area of a circular sector is A = (1/2)r²θ Step 2: Substitute the given values: 24π = (1/2)(8²)θ Step 3: Simplify: 24π = (1/2)(64)θ Step 4: Calculate: 24π = 32θ Step 5: Solve for θ: θ = 24π/32 Step 6: Simplify the fraction: θ = 3π/4 The central angle is 3π/4 radians.