A circular sedimentation basin of radius 8.0 ft is partitioned by a central angle of 143°. 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

   $$s = 8.0 \times 2.4958 = 19.97 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(8.0)^{2}(2.4958) = 79.87 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(8.0)^{2}(2.4958 - 0.6018) = 60.61 ft^{2}$$

Why the other options are there

• 4,576 ft² (degrees used directly)
• 47.87 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 14.0 ft is partitioned by a central angle of 83°. 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

   $$s = 14.0 \times 1.4486 = 20.28 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(14.0)^{2}(1.4486) = 142.0 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(14.0)^{2}(1.4486 - 0.9925) = 44.70 ft^{2}$$

Why the other options are there

• 8,134 ft² (degrees used directly)
• 43.97 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 10.5 ft is partitioned by a central angle of 150°. 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

   $$s = 10.5 \times 2.6180 = 27.49 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(10.5)^{2}(2.6180) = 144.3 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(10.5)^{2}(2.6180 - 0.5000) = 116.8 ft^{2}$$

Why the other options are there

• 8,269 ft² (degrees used directly)
• 89.19 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 7.0 ft is partitioned by a central angle of 100°. 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

   $$s = 7.0 \times 1.7453 = 12.22 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.0)^{2}(1.7453) = 42.76 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.0)^{2}(1.7453 - 0.9848) = 18.63 ft^{2}$$

Why the other options are there

• 2,450 ft² (degrees used directly)
• 18.26 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 7.5 ft is partitioned by a central angle of 86°. 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

   $$s = 7.5 \times 1.5010 = 11.26 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.5)^{2}(1.5010) = 42.22 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.5)^{2}(1.5010 - 0.9976) = 14.16 ft^{2}$$

Why the other options are there

• 2,419 ft² (degrees used directly)
• 14.09 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 13.5 ft is partitioned by a central angle of 65°. 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

   $$s = 13.5 \times 1.1345 = 15.32 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.5)^{2}(1.1345) = 103.4 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.5)^{2}(1.1345 - 0.9063) = 20.79 ft^{2}$$

Why the other options are there

• 5,923 ft² (degrees used directly)
• 12.25 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 11.5 ft is partitioned by a central angle of 44°. 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

   $$s = 11.5 \times 0.7679 = 8.83 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(11.5)^{2}(0.7679) = 50.78 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(11.5)^{2}(0.7679 - 0.6947) = 4.85 ft^{2}$$

Why the other options are there

• 2,910 ft² (degrees used directly)
• -15.34 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 8.5 ft is partitioned by a central angle of 42°. 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

   $$s = 8.5 \times 0.7330 = 6.23 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(8.5)^{2}(0.7330) = 26.48 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(8.5)^{2}(0.7330 - 0.6691) = 2.31 ft^{2}$$

Why the other options are there

• 1,517 ft² (degrees used directly)
• -9.64 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 6.5 ft is partitioned by a central angle of 139°. 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

   $$s = 6.5 \times 2.4260 = 15.77 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.5)^{2}(2.4260) = 51.25 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(6.5)^{2}(2.4260 - 0.6561) = 37.39 ft^{2}$$

Why the other options are there

• 2,936 ft² (degrees used directly)
• 30.12 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone

A circular sedimentation basin of radius 7.0 ft is partitioned by a central angle of 120°. 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

   $$s = 7.0 \times 2.0944 = 14.66 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.0)^{2}(2.0944) = 51.31 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.0)^{2}(2.0944 - 0.8660) = 30.10 ft^{2}$$

Why the other options are there

• 2,940 ft² (degrees used directly)
• 26.81 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cone