Arc Lengths and Sector Area.pdf - Siera Caulton Analytic Geometry ... - Free Printable

I'll solve the problem in this image, which appears to be a worksheet on "Arc Length and Sector Area." I'll go through each question step by step.

First, let me look closely at the worksheet to understand what's being asked and what information is provided.

The worksheet has 8 problems, each with a circle diagram showing a central angle and sometimes an arc length or sector area. The problems ask to find either the arc length or the sector area, rounding answers to the nearest tenth.

Let me start with problem 1:
- It shows a circle with a central angle of 90° (a quarter circle)
- The radius is given as 12 units
- We need to find the arc length

For arc length, the formula is: Arc Length = (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius.

For problem 1: Arc Length = (90/360) × 2π × 12 = (1/4) × 24π = 6π ≈ 18.8 units

Let me continue with problem 2:
- Central angle is 150°
- Radius is 8 units
- Need to find arc length

Arc Length = (150/360) × 2π × 8 = (5/12) × 16π = (80/12)π = (20/3)π ≈ 20.9 units

Problem 3:
- Central angle is 90°
- Radius is 7 units
- Need to find arc length

Arc Length = (90/360) × 2π × 7 = (1/4) × 14π = 3.5π ≈ 11.0 units

Problem 4:
- Central angle is 60°
- Radius is 13 units
- Need to find arc length

Arc Length = (60/360) × 2π × 13 = (1/6) × 26π = (26/6)π = (13/3)π ≈ 13.6 units

Problem 5:
- This asks for sector area
- Central angle is 120°
- Radius is 10 units

For sector area, the formula is: Sector Area = (θ/360) × πr²

Sector Area = (120/360) × π × 10² = (1/3) × 100π = (100/3)π ≈ 104.7 square units

Problem 6:
- Sector area
- Central angle is 150°
- Radius is 11 units

Sector Area = (150/360) × π × 11² = (5/12) × 121π = (605/12)π ≈ 158.3 square units

Problem 7:
- Sector area
- Central angle is 270°
- Radius is 14 units

Sector Area = (270/360) × π × 14² = (3/4) × 196π = 147π ≈ 461.8 square units

Problem 8:
- Sector area
- Central angle is 60°
- Radius is 10 units

Sector Area = (60/360) × π × 10² = (1/6) × 100π = (100/6)π ≈ 52.4 square units

Now I'll provide the final answers for each problem:

1. Arc Length ≈ 18.8 units
2. Arc Length ≈ 20.9 units
3. Arc Length ≈ 11.0 units
4. Arc Length ≈ 13.6 units
5. Sector Area ≈ 104.7 square units
6. Sector Area ≈ 158.3 square units
7. Sector Area ≈ 461.8 square units
8. Sector Area ≈ 52.4 square units