A circular sedimentation basin of radius 8.5 ft is partitioned by a central angle of 110°. 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

   $$s = 8.5 \times 1.9199 = 16.32 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(8.5)^{2}(1.9199) = 69.36 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(8.5)^{2}(1.9199 - 0.9397) = 35.41 ft^{2}$$

Why the other options are there

• 3,974 ft² (degrees used directly)
• 33.23 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 9.5 ft is partitioned by a central angle of 138°. 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

   $$s = 9.5 \times 2.4086 = 22.88 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(9.5)^{2}(2.4086) = 108.7 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(9.5)^{2}(2.4086 - 0.6691) = 78.49 ft^{2}$$

Why the other options are there

• 6,227 ft² (degrees used directly)
• 63.56 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 10.5 ft is partitioned by a central angle of 53°. 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

   $$s = 10.5 \times 0.9250 = 9.71 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(10.5)^{2}(0.9250) = 50.99 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(10.5)^{2}(0.9250 - 0.7986) = 6.97 ft^{2}$$

Why the other options are there

• 2,922 ft² (degrees used directly)
• -4.13 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 4.0 ft is partitioned by a central angle of 153°. 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

   $$s = 4.0 \times 2.6704 = 10.68 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(4.0)^{2}(2.6704) = 21.36 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(4.0)^{2}(2.6704 - 0.4540) = 17.73 ft^{2}$$

Why the other options are there

• 1,224 ft² (degrees used directly)
• 13.36 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 7.5 ft is partitioned by a central angle of 135°. 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

   $$s = 7.5 \times 2.3562 = 17.67 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.5)^{2}(2.3562) = 66.27 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.5)^{2}(2.3562 - 0.7071) = 46.38 ft^{2}$$

Why the other options are there

• 3,797 ft² (degrees used directly)
• 38.14 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 7.0 ft is partitioned by a central angle of 147°. 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

   $$s = 7.0 \times 2.5656 = 17.96 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.0)^{2}(2.5656) = 62.86 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.0)^{2}(2.5656 - 0.5446) = 49.51 ft^{2}$$

Why the other options are there

• 3,602 ft² (degrees used directly)
• 38.36 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 8.5 ft is partitioned by a central angle of 53°. 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

   $$s = 8.5 \times 0.9250 = 7.86 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(8.5)^{2}(0.9250) = 33.42 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(8.5)^{2}(0.9250 - 0.7986) = 4.57 ft^{2}$$

Why the other options are there

• 1,915 ft² (degrees used directly)
• -2.71 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 9.0 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

   $$s = 9.0 \times 1.4486 = 13.04 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(9.0)^{2}(1.4486) = 58.67 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(9.0)^{2}(1.4486 - 0.9925) = 18.47 ft^{2}$$

Why the other options are there

• 3,362 ft² (degrees used directly)
• 18.17 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 6.0 ft is partitioned by a central angle of 117°. 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

   $$s = 6.0 \times 2.0420 = 12.25 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.0)^{2}(2.0420) = 36.76 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(6.0)^{2}(2.0420 - 0.8910) = 20.72 ft^{2}$$

Why the other options are there

• 2,106 ft² (degrees used directly)
• 18.76 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution

A circular sedimentation basin of radius 11.0 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

   $$s = 11.0 \times 0.7330 = 8.06 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(11.0)^{2}(0.7330) = 44.35 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(11.0)^{2}(0.7330 - 0.6691) = 3.87 ft^{2}$$

Why the other options are there

• 2,541 ft² (degrees used directly)
• -16.15 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Paraboloid of Revolution