A circular sedimentation basin of radius 8.5 ft is partitioned by a central angle of 151°. 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

   $$s = 8.5 \times 2.6354 = 22.40 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(8.5)^{2}(2.6354) = 95.21 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(8.5)^{2}(2.6354 - 0.4848) = 77.69 ft^{2}$$

Why the other options are there

• 5,455 ft² (degrees used directly)
• 59.08 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 9.0 ft is partitioned by a central angle of 112°. 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

   $$s = 9.0 \times 1.9548 = 17.59 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(9.0)^{2}(1.9548) = 79.17 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(9.0)^{2}(1.9548 - 0.9272) = 41.62 ft^{2}$$

Why the other options are there

• 4,536 ft² (degrees used directly)
• 38.67 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 10.5 ft is partitioned by a central angle of 152°. 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

   $$s = 10.5 \times 2.6529 = 27.86 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(10.5)^{2}(2.6529) = 146.2 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(10.5)^{2}(2.6529 - 0.4695) = 120.4 ft^{2}$$

Why the other options are there

• 8,379 ft² (degrees used directly)
• 91.12 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 7.5 ft is partitioned by a central angle of 70°. 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

   $$s = 7.5 \times 1.2217 = 9.16 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.5)^{2}(1.2217) = 34.36 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.5)^{2}(1.2217 - 0.9397) = 7.93 ft^{2}$$

Why the other options are there

• 1,969 ft² (degrees used directly)
• 6.24 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 3.5 ft is partitioned by a central angle of 55°. 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

   $$s = 3.5 \times 0.9599 = 3.36 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(3.5)^{2}(0.9599) = 5.88 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(3.5)^{2}(0.9599 - 0.8192) = 0.86 ft^{2}$$

Why the other options are there

• 336.9 ft² (degrees used directly)
• -0.25 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 4.5 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

   $$s = 4.5 \times 2.5656 = 11.55 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(4.5)^{2}(2.5656) = 25.98 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(4.5)^{2}(2.5656 - 0.5446) = 20.46 ft^{2}$$

Why the other options are there

• 1,488 ft² (degrees used directly)
• 15.85 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 7.0 ft is partitioned by a central angle of 124°. 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

   $$s = 7.0 \times 2.1642 = 15.15 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.0)^{2}(2.1642) = 53.02 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.0)^{2}(2.1642 - 0.8290) = 32.71 ft^{2}$$

Why the other options are there

• 3,038 ft² (degrees used directly)
• 28.52 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 4.0 ft is partitioned by a central angle of 41°. 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

   $$s = 4.0 \times 0.7156 = 2.86 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(4.0)^{2}(0.7156) = 5.72 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(4.0)^{2}(0.7156 - 0.6561) = 0.48 ft^{2}$$

Why the other options are there

• 328.0 ft² (degrees used directly)
• -2.28 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 10.0 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

   $$s = 10.0 \times 2.7751 = 27.75 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(10.0)^{2}(2.7751) = 138.8 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(10.0)^{2}(2.7751 - 0.3584) = 120.8 ft^{2}$$

Why the other options are there

• 7,950 ft² (degrees used directly)
• 88.75 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment

A circular sedimentation basin of radius 7.5 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

   $$s = 7.5 \times 2.0420 = 15.32 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.5)^{2}(2.0420) = 57.43 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(7.5)^{2}(2.0420 - 0.8910) = 32.37 ft^{2}$$

Why the other options are there

• 3,291 ft² (degrees used directly)
• 29.31 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Circular Segment