Practice worksheet for calculating arc lengths and sector areas in circles, featuring nine problems with diagrams and measurements.

Let me solve all 9 problems on this Arc Length & Sector Area practice worksheet.

Key Formulas:
- Arc Length: L = (θ/360) × 2πr = (θ/360) × πd
- Sector Area: A = (θ/360) × πr²

Let me work through each problem carefully.

Problem 1: Find the length of the bolded arc. r = 3 mi, angle shown is 108°, but the bolded arc appears to be the major arc (the larger portion). Let me look again - the bolded arc is on the left side which looks like it's the major arc. The angle marked is 108° for the minor sector. So the bolded arc corresponds to 360° - 108° = 252°.

Arc Length = (252/360) × 2π(3) = (252/360) × 6π = 0.7 × 6π = 4.2π ≈ 13.19 mi

Wait, let me reconsider. Looking at the image description again - "108°" with radius "3 mi" and the bolded arc. The bolded arc could be either the minor or major arc. In typical worksheets, if the angle shown is 108° and the bolded arc is the one NOT corresponding to that angle, then it's the major arc = 360 - 108 = 252°.

Actually, looking more carefully at typical worksheet conventions - the bolded arc usually corresponds to the angle shown unless it's clearly the other way. Let me assume the bolded arc is the major arc (252°) since the 108° angle is shown for the unbolded part.

L = (252/360) × 2π(3) = (7/10) × 6π = 4.2π ≈ 13.19 mi

Hmm, but actually I should reconsider. Let me just compute both possibilities and see which makes more sense as a "clean" answer.

If θ = 108°: L = (108/360) × 2π(3) = 0.3 × 6π = 1.8π ≈ 5.65 mi
If θ = 252°: L = (252/360) × 2π(3) = 0.7 × 6π = 4.2π ≈ 13.19 mi

Looking at the image description again - the bolded arc appears to be on the LEFT side of the circle, while the 108° angle is shown. If the bolded arc is the larger portion (major arc), then it's 252°.

I'll go with the major arc interpretation: L = 4.2π ≈ 13.19 mi

Actually, let me reconsider once more. In many worksheets, the bolded arc IS the arc corresponding to the given angle. Let me check problem 3 which has 143° and radius 6 in - if the bolded arc corresponds to 143°, that would be a specific calculation.

For problem 1, I'll assume the bolded arc is the MAJOR arc (the larger one not covered by the 108° angle), so θ = 360 - 108 = 252°.

L = (252/360)(2π)(3) = (0.7)(6π) = 4.2π ≈ 13.19 mi

Problem 2: Find the area of the shaded sector. r = 8 ft, θ = 32°
A = (32/360) × π(8)² = (32/360) × 64π = (2048/360)π = (256/45)π ≈ 17.87 ft²

Problem 3: Find the length of the bolded arc. r = 6 in, angle = 143°. The bolded arc appears to be the major arc based on the description.
If major arc: θ = 360 - 143 = 217°
L = (217/360) × 2π(6) = (217/360) × 12π = (217/30)π ≈ 22.72 in

If the bolded arc is the minor arc (143°):
L = (143/360) × 12π = (143/30)π ≈ 14.97 in

Looking at the image - "the bolded arc" with 143° shown. I'll assume the bolded arc is the major arc = 217°.
L = (217/360)(12π) = 7.233π ≈ 22.72 in

Hmm, actually I need to be more careful here. Let me reconsider the convention. In most worksheets I've seen, when an angle is given and an arc is bolded, the bolded arc typically corresponds to that angle UNLESS the diagram clearly shows otherwise.

Let me re-examine: For problem 1, the 108° angle is shown and the bolded arc is described as being on the left. If the 108° is the angle of the sector on the right (unbolded), then the bolded arc on the left would be 360-108 = 252°.

For problem 3, similarly, if 143° is shown for the unbolded sector, the bolded arc would be 360-143 = 217°.

I'll proceed with this interpretation.

Problem 4: Find the area of the shaded sector. r = 12 cm, θ = 204°
A = (204/360) × π(12)² = (204/360) × 144π = (204 × 144/360)π = (29376/360)π = 81.6π ≈ 256.35 cm²

Simplifying: 204/360 = 17/30
A = (17/30)(144π) = (17 × 144/30)π = (2448/30)π = 81.6π ≈ 256.35 cm²

Problem 5: Find the length of the bolded arc. d = 30 m, so r = 15 m, θ = 47°. Bolded arc is likely the major arc = 360 - 47 = 313°.
L = (313/360) × 2π(15) = (313/360) × 30π = (313/12)π ≈ 81.94 m

If minor arc (47°):
L = (47/360) × 30π = (47/12)π ≈ 12.30 m

I'll go with major arc: L = (313/12)π ≈ 81.94 m

Problem 6: Find the area of the shaded sector. d = 17 mi, so r = 8.5 mi, θ = 126°
A = (126/360) × π(8.5)² = (126/360) × 72.25π = (126 × 72.25/360)π = (9103.5/360)π = 25.2875π ≈ 79.44 mi²

Simplifying: 126/360 = 7/20
A = (7/20)(72.25π) = (505.75/20)π = 25.2875π ≈ 79.44 mi²

Problem 7: Find the length of an arc subtended by a 70° central angle in a circle with a radius of 10 cm.
L = (70/360) × 2π(10) = (70/360) × 20π = (1400/360)π = (35/9)π ≈ 12.22 cm

Problem 8: Find the area of a sector formed by a 105° central angle in a circle with a radius of 5 in.
A = (105/360) × π(5)² = (105/360) × 25π = (2625/360)π = (175/24)π ≈ 22.91 in²

Problem 9: What is the length of an arc created by a 280° central angle in a circle with a diameter of 100 m?
r = 50 m
L = (280/360) × 2π(50) = (280/360) × 100π = (28000/360)π = (700/9)π ≈ 244.35 m

Let me verify my calculations using Python.