A circular sedimentation basin of radius 9.5 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

   $$s = 9.5 \times 2.6529 = 25.20 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(9.5)^{2}(2.6529) = 119.7 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(9.5)^{2}(2.6529 - 0.4695) = 98.53 ft^{2}$$

Why the other options are there

• 6,859 ft² (degrees used directly)
• 74.59 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 12.0 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

   $$s = 12.0 \times 1.4835 = 17.80 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(12.0)^{2}(1.4835) = 106.8 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(12.0)^{2}(1.4835 - 0.9962) = 35.09 ft^{2}$$

Why the other options are there

• 6,120 ft² (degrees used directly)
• 34.81 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 13.0 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

   $$s = 13.0 \times 1.5010 = 19.51 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.0)^{2}(1.5010) = 126.8 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.0)^{2}(1.5010 - 0.9976) = 42.54 ft^{2}$$

Why the other options are there

• 7,267 ft² (degrees used directly)
• 42.33 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 6.0 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

   $$s = 6.0 \times 2.5482 = 15.29 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.0)^{2}(2.5482) = 45.87 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(6.0)^{2}(2.5482 - 0.5592) = 35.80 ft^{2}$$

Why the other options are there

• 2,628 ft² (degrees used directly)
• 27.87 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 13.5 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

   $$s = 13.5 \times 1.2217 = 16.49 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.5)^{2}(1.2217) = 111.3 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.5)^{2}(1.2217 - 0.9397) = 25.70 ft^{2}$$

Why the other options are there

• 6,379 ft² (degrees used directly)
• 20.21 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 6.5 ft is partitioned by a central angle of 58°. 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

   $$s = 6.5 \times 1.0123 = 6.58 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.5)^{2}(1.0123) = 21.38 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(6.5)^{2}(1.0123 - 0.8480) = 3.47 ft^{2}$$

Why the other options are there

• 1,225 ft² (degrees used directly)
• 0.26 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 4.0 ft is partitioned by a central angle of 40°. 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

   $$s = 4.0 \times 0.6981 = 2.79 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(4.0)^{2}(0.6981) = 5.59 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(4.0)^{2}(0.6981 - 0.6428) = 0.44 ft^{2}$$

Why the other options are there

• 320.0 ft² (degrees used directly)
• -2.41 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 13.0 ft is partitioned by a central angle of 144°. 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

   $$s = 13.0 \times 2.5133 = 32.67 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(13.0)^{2}(2.5133) = 212.4 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(13.0)^{2}(2.5133 - 0.5878) = 162.7 ft^{2}$$

Why the other options are there

• 12,168 ft² (degrees used directly)
• 127.9 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 6.0 ft is partitioned by a central angle of 48°. 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

   $$s = 6.0 \times 0.8378 = 5.03 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.0)^{2}(0.8378) = 15.08 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(6.0)^{2}(0.8378 - 0.7431) = 1.70 ft^{2}$$

Why the other options are there

• 864.0 ft² (degrees used directly)
• -2.92 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid

A circular sedimentation basin of radius 5.5 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

   $$s = 5.5 \times 2.0944 = 11.52 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(5.5)^{2}(2.0944) = 31.68 ft^{2}$$

5. Formula

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

6. Substituting

   $$A_segment = ½(5.5)^{2}(2.0944 - 0.8660) = 18.58 ft^{2}$$

Why the other options are there

• 1,815 ft² (degrees used directly)
• 16.55 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Prismoid