A circular sedimentation basin of radius 5.0 ft is partitioned by a central angle of 132°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 5.0 \times 2.3038 = 11.52 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(5.0)^{2}(2.3038) = 28.80 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(5.0)^{2}(2.3038 - 0.7431) = 19.51 ft^{2}$$

Why the other options are there

• 1,650 ft² (degrees used directly)
• 16.30 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 7.5 ft is partitioned by a central angle of 146°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 7.5 \times 2.5482 = 19.11 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.5)^{2}(2.5482) = 71.67 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(7.5)^{2}(2.5482 - 0.5592) = 55.94 ft^{2}$$

Why the other options are there

• 4,106 ft² (degrees used directly)
• 43.54 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 10.0 ft is partitioned by a central angle of 60°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 10.0 \times 1.0472 = 10.47 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(10.0)^{2}(1.0472) = 52.36 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(10.0)^{2}(1.0472 - 0.8660) = 9.06 ft^{2}$$

Why the other options are there

• 3,000 ft² (degrees used directly)
• 2.36 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 13.5 ft is partitioned by a central angle of 88°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 13.5 \times 1.5359 = 20.73 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.5)^{2}(1.5359) = 140.0 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(13.5)^{2}(1.5359 - 0.9994) = 48.89 ft^{2}$$

Why the other options are there

• 8,019 ft² (degrees used directly)
• 48.83 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 9.0 ft is partitioned by a central angle of 77°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 9.0 \times 1.3439 = 12.10 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(9.0)^{2}(1.3439) = 54.43 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(9.0)^{2}(1.3439 - 0.9744) = 14.97 ft^{2}$$

Why the other options are there

• 3,119 ft² (degrees used directly)
• 13.93 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 6.5 ft is partitioned by a central angle of 106°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 6.5 \times 1.8500 = 12.03 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.5)^{2}(1.8500) = 39.08 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(6.5)^{2}(1.8500 - 0.9613) = 18.78 ft^{2}$$

Why the other options are there

• 2,239 ft² (degrees used directly)
• 17.96 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 12.0 ft is partitioned by a central angle of 159°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 12.0 \times 2.7751 = 33.30 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(12.0)^{2}(2.7751) = 199.8 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(12.0)^{2}(2.7751 - 0.3584) = 174.0 ft^{2}$$

Why the other options are there

• 11,448 ft² (degrees used directly)
• 127.8 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 6.0 ft is partitioned by a central angle of 81°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 6.0 \times 1.4137 = 8.48 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.0)^{2}(1.4137) = 25.45 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(6.0)^{2}(1.4137 - 0.9877) = 7.67 ft^{2}$$

Why the other options are there

• 1,458 ft² (degrees used directly)
• 7.45 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 11.0 ft is partitioned by a central angle of 84°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 11.0 \times 1.4661 = 16.13 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(11.0)^{2}(1.4661) = 88.70 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(11.0)^{2}(1.4661 - 0.9945) = 28.53 ft^{2}$$

Why the other options are there

• 5,082 ft² (degrees used directly)
• 28.20 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector

A circular sedimentation basin of radius 11.0 ft is partitioned by a central angle of 136°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 11.0 \times 2.3736 = 26.11 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(11.0)^{2}(2.3736) = 143.6 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(11.0)^{2}(2.3736 - 0.6947) = 101.6 ft^{2}$$

Why the other options are there

• 8,228 ft² (degrees used directly)
• 83.11 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Sector