A circular sedimentation basin of radius 5.0 ft is partitioned by a central angle of 107°. 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

   $$s = 5.0 \times 1.8675 = 9.34 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(5.0)^{2}(1.8675) = 23.34 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(5.0)^{2}(1.8675 - 0.9563) = 11.39 ft^{2}$$

Why the other options are there

• 1,338 ft² (degrees used directly)
• 10.84 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 13.0 ft is partitioned by a central angle of 45°. 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

   $$s = 13.0 \times 0.7854 = 10.21 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.0)^{2}(0.7854) = 66.37 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.0)^{2}(0.7854 - 0.7071) = 6.62 ft^{2}$$

Why the other options are there

• 3,803 ft² (degrees used directly)
• -18.13 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 12.5 ft is partitioned by a central angle of 85°. 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

   $$s = 12.5 \times 1.4835 = 18.54 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(12.5)^{2}(1.4835) = 115.9 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(12.5)^{2}(1.4835 - 0.9962) = 38.07 ft^{2}$$

Why the other options are there

• 6,641 ft² (degrees used directly)
• 37.78 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 3.5 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

   $$s = 3.5 \times 2.3038 = 8.06 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(3.5)^{2}(2.3038) = 14.11 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(3.5)^{2}(2.3038 - 0.7431) = 9.56 ft^{2}$$

Why the other options are there

• 808.5 ft² (degrees used directly)
• 7.99 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 11.0 ft is partitioned by a central angle of 61°. 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

   $$s = 11.0 \times 1.0647 = 11.71 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(11.0)^{2}(1.0647) = 64.41 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(11.0)^{2}(1.0647 - 0.8746) = 11.50 ft^{2}$$

Why the other options are there

• 3,691 ft² (degrees used directly)
• 3.91 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 14.0 ft is partitioned by a central angle of 91°. 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

   $$s = 14.0 \times 1.5882 = 22.24 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(14.0)^{2}(1.5882) = 155.6 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(14.0)^{2}(1.5882 - 0.9998) = 57.66 ft^{2}$$

Why the other options are there

• 8,918 ft² (degrees used directly)
• 57.65 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 13.5 ft is partitioned by a central angle of 148°. 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

   $$s = 13.5 \times 2.5831 = 34.87 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.5)^{2}(2.5831) = 235.4 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.5)^{2}(2.5831 - 0.5299) = 187.1 ft^{2}$$

Why the other options are there

• 13,487 ft² (degrees used directly)
• 144.3 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 12.5 ft is partitioned by a central angle of 102°. 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

   $$s = 12.5 \times 1.7802 = 22.25 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(12.5)^{2}(1.7802) = 139.1 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(12.5)^{2}(1.7802 - 0.9781) = 62.66 ft^{2}$$

Why the other options are there

• 7,969 ft² (degrees used directly)
• 60.96 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 10.0 ft is partitioned by a central angle of 125°. 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

   $$s = 10.0 \times 2.1817 = 21.82 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(10.0)^{2}(2.1817) = 109.1 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(10.0)^{2}(2.1817 - 0.8192) = 68.13 ft^{2}$$

Why the other options are there

• 6,250 ft² (degrees used directly)
• 59.08 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)

A circular sedimentation basin of radius 13.0 ft is partitioned by a central angle of 68°. 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

   $$s = 13.0 \times 1.1868 = 15.43 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.0)^{2}(1.1868) = 100.3 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.0)^{2}(1.1868 - 0.9272) = 21.94 ft^{2}$$

Why the other options are there

• 5,746 ft² (degrees used directly)
• 15.79 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Regular Polygon (n equal sides)