A circle with center $O$ has a radius of 10 cm. If a chord $AB$ subtends an angle of $90^\circ$ at the center, what is the area of the segment $APB$? Use $\pi = 3.14$.

Correct Option: A. Solution: The area of the sector $OAPB$ is given by $$\frac{90}{360} \times \pi \times 10^2 = 25\pi = 78.5\text{ cm}^2$$. The area of triangle $OAB$ is $$\frac{1}{2} \times 10 \times 10 \times \sin(90^\circ) = 50\text{ cm}^2$$. Thus, the area of the segment $APB$ is $$78.5 - 50 = 28.5\text{ cm}^2$$.

Areas Related to Circles

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Summary

Chapter Summary: Areas Related to Circles

Key Points

Length of an Arc:
- Formula: $L = \frac{\Theta}{360} \times 2\pi r$
Area of a Sector:
- Formula: $A = \frac{\Theta}{360} \times \pi r^2$
Area of a Segment:
- Formula: $\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}$

Definitions

Sector: Portion of a circular region enclosed by two radii and the corresponding arc.
Segment: Portion of a circular region enclosed between a chord and the corresponding arc.
Minor Sector: Smaller sector formed by the angle at the center.
Major Sector: Larger sector formed by the angle at the center.
Minor Segment: Smaller segment formed by the chord.
Major Segment: Larger segment formed by the chord.

Examples

Example 1: Area of sector with radius 4 cm and angle 30°:
- Area = 4.19 cm² (approx.)
- Major Sector Area = 46.1 cm² (approx.)
Example 2: Area of segment AYB with radius 21 cm and angle 120°:
- Area = 462 cm² (for sector) - Area of triangle AOB.

Learning Objectives

Understand the concepts of sectors and segments of a circle.
Calculate the length of an arc of a sector given the radius and angle.
Determine the area of a sector using the formula:
- Area of the sector = $\frac{\Theta}{360} \times \pi r^2$
Compute the area of a segment of a circle using:
- Area of the segment = Area of the sector - Area of the triangle formed by the radii and the chord.
Apply the formulas to solve real-world problems involving circles.

Detailed Notes

Areas Related to Circles

Key Concepts

Sector of a Circle: The portion of the circular region enclosed by two radii and the corresponding arc.
Segment of a Circle: The portion of the circular region enclosed between a chord and the corresponding arc.

Formulas

Length of an Arc:

$L = \frac{\Theta}{360} \times 2\pi r$

Where:
- $L$ = Length of the arc
- $r$ = Radius of the circle
- $\Theta$ = Angle in degrees
Area of a Sector:

$A = \frac{\Theta}{360} \times \pi r^2$

Where:
- $A$ = Area of the sector
- $r$ = Radius of the circle
- $\Theta$ = Angle in degrees
Area of a Segment:

$A_{segment} = A_{sector} - A_{triangle}$

Where:
- $A_{segment}$ = Area of the segment
- $A_{sector}$ = Area of the corresponding sector
- $A_{triangle}$ = Area of the corresponding triangle

Examples

Example 1: Area of a Sector

Given: Radius = 4 cm, Angle = 30°
Calculation:
- Area of the sector = $\frac{30}{360} \times \pi \times 4^2 \approx 4.19 \text{ cm}^2$
- Area of the major sector = $\pi r^2 - \text{Area of sector} \approx 46.1 \text{ cm}^2$

Example 2: Area of a Segment

Given: Radius = 21 cm, Angle = 120°
Calculation:
- Area of the sector = $\frac{120}{360} \times \frac{22}{7} \times 21^2 = 462 \text{ cm}^2$
- Area of the segment = Area of sector - Area of triangle

Diagrams

Fig. 11.1: Circle with labeled sectors (major and minor).
Fig. 11.2: Circle with labeled segments (major and minor).
Fig. 11.5: Circle with a sector and a central angle.
Fig. 11.6: Circle with a segment and a central angle.

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Sector and Segment Definitions: Students often confuse sectors and segments of a circle. Remember that a sector is the area enclosed by two radii and the arc, while a segment is the area enclosed by a chord and the arc.
Incorrect Formula Application: Ensure you apply the correct formulas for the area of a sector and segment. The area of a sector is given by $\frac{\Theta}{360} \times \pi r^2$ and the area of a segment is the area of the sector minus the area of the triangle formed by the radii and the chord.
Neglecting Units: Always include units in your calculations. For example, if the radius is in cm, the area will be in cm².

Tips for Success

Draw Diagrams: Visual aids can help clarify problems involving sectors and segments. Label all parts of the diagram to avoid confusion.
Practice with Examples: Work through examples that require calculating areas of sectors and segments to reinforce understanding.
Check Your Work: After solving a problem, revisit the formulas used and ensure all calculations are correct, especially when converting angles and applying the formulas.

Practice & Assessment

Multiple Choice Questions

21.5 cm²

23.5 cm²

25.5 cm²

27.5 cm²

Correct Answer: A

Solution:

The area of the sector

OAPB

is given by

\frac{90}{360} \times \pi \times 10^2 = 25\pi = 78.5\text{ cm}^2

. The area of triangle

OAB

\frac{1}{2} \times 10 \times 10 \times \sin(90^\circ) = 50\text{ cm}^2

. Thus, the area of the segment

APB

78.5 - 50 = 28.5\text{ cm}^2

43.56 cm²

45.56 cm²

47.56 cm²

49.56 cm²

Correct Answer: B

Solution:

The area of the major segment is the area of the circle minus the area of the minor segment. The area of the minor segment is the area of the sector minus the area of the triangle. The area of the sector is

\frac{120}{360} \times \pi \times 6^2 = 37.68 \text{ cm}^2

. The area of the triangle is

\frac{1}{2} \times 6 \times 6 \times \sin(120°) = 15.59 \text{ cm}^2

. Therefore, the area of the minor segment is

37.68 - 15.59 = 22.09 \text{ cm}^2

. The area of the circle is

\pi \times 6^2 = 113.04 \text{ cm}^2

. The area of the major segment is

113.04 - 22.09 = 90.95 \text{ cm}^2

\frac{\pi r}{2}

\frac{\pi r}{4}

\pi r

2\pi r

Correct Answer: A

Solution:

The length of an arc is given by the formula:

\frac{\Theta}{360} \times 2\pi r

. Substituting the given values, we have:

\frac{90}{360} \times 2\pi r = \frac{\pi r}{2}

4π cm

5π cm

6π cm

7π cm

Correct Answer: B

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Substituting the given values,

\frac{72}{360} \times 2\pi \times 10 = 5\pi

cm.

80\pi cm²

60\pi cm²

40\pi cm²

20\pi cm²

Correct Answer: A

Solution:

The area of the major segment is the area of the circle minus the area of the minor segment:

100\pi - 20\pi = 80\pi

cm².

31.4 cm

47.1 cm

62.8 cm

78.5 cm

Correct Answer: B

Solution:

The perimeter of a sector is the sum of the two radii and the arc length. Arc length

= \frac{60}{360} \times 2 \times 3.14 \times 15 = 15.7 \text{ cm}

. Perimeter

= 15 + 15 + 15.7 = 47.1 \text{ cm}

The area of the sector quadruples.

The area of the sector doubles.

The area of the sector remains the same.

The area of the sector triples.

Correct Answer: A

Solution:

The area of a sector is given by

\frac{\Theta}{360} \times \pi r^2

. If the radius is doubled, the area becomes

\frac{\Theta}{360} \times \pi (2r)^2 = 4 \times \frac{\Theta}{360} \times \pi r^2

, thus quadrupling.

3π cm

6π cm

9π cm

12π cm

Correct Answer: B

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Substituting the values,

\frac{45}{360} \times 2\pi \times 12 = 6\pi

cm.

90°

180°

120°

60°

Correct Answer: A

Solution:

Using the formula for the area of a sector,

\frac{\Theta}{360} \times \pi r^2 = 50\pi

. Solving for

\Theta

gives

\Theta = 90°

10 cm

12 cm

14 cm

16 cm

Correct Answer: B

Solution:

Using the Pythagorean theorem in the triangle formed by the radius, the distance from the center to the chord, and half the chord length, we have

8^2 = 6^2 + (\frac{c}{2})^2

. Solving for

c

, we get

c = 12

cm.

11 cm

10 cm

12 cm

13 cm

Correct Answer: A

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Substituting the values, we get

\frac{90}{360} \times 2 \times 3.14 \times 7 = 11 \text{ cm}

10 cm²

5 cm²

15 cm²

20 cm²

Correct Answer: A

Solution:

The area of the segment is the area of the sector minus the area of the triangle. Thus, the area of the triangle is

30 - 20 = 10 \text{ cm}^2

90°

120°

180°

240°

Correct Answer: A

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Solving

\frac{\theta}{360} \times 2\pi \times 8 = 4\pi

, we find

\theta = 90°

81π cm²

81π/4 cm²

243π/4 cm²

162π cm²

Correct Answer: C

Solution:

The area of the major sector is

\pi r^2 - \frac{135}{360} \times \pi r^2

. Substituting the given values,

81\pi - \frac{135}{360} \times 81\pi = \frac{243\pi}{4}

cm².

47.12 cm²

78.54 cm²

94.25 cm²

117.81 cm²

Correct Answer: D

Solution:

The area of a sector is given by

\frac{\theta}{360} \times \pi r^2

. Substituting the values, we get

\frac{120}{360} \times \pi \times 15^2 = 117.81 \text{ cm}^2

282.6 cm²

314 cm²

235.5 cm²

157 cm²

Correct Answer: C

Solution:

The area of the major sector is

\pi r^2 - \frac{\theta}{360} \times \pi r^2

. Substituting the values, we get

3.14 \times 10^2 - \frac{45}{360} \times 3.14 \times 10^2 = 235.5 \text{ cm}^2

114.6°

120°

90°

100°

Correct Answer: A

Solution:

The area of a sector is given by

\frac{\theta}{360} \times \pi r^2

. Solving for

\theta

, we have

\theta = \frac{360 \times 50}{\pi \times 5^2} = 114.6°

9.42 cm

18.84 cm

28.26 cm

37.68 cm

Correct Answer: A

Solution:

The length of an arc is

\frac{\theta}{360} \times 2\pi r

. Substituting

\theta = 60°

and

r = 9

cm, we get

\frac{60}{360} \times 2\pi \times 9 = 9.42

cm.

45°

90°

180°

270°

Correct Answer: B

Solution:

Using the formula for the area of a sector,

\frac{\theta}{360} \times \pi \times 10^2 = 25\pi

. Solving for

\theta

gives

\theta = 90°

4 cm

5 cm

6 cm

7 cm

Correct Answer: B

Solution:

Let

d

be the distance from the center to the chord. By the Pythagorean theorem in the right triangle formed by the radius, half the chord, and the perpendicular distance from the center to the chord, we have:

8^2 = 5^2 + d^2.

Therefore,

64 = 25 + d^2 \Rightarrow d^2 = 39 \Rightarrow d = \sqrt{39} \approx 6.24 \text{ cm}.

Rounding to the nearest whole number,

d

is approximately 6 cm.

13.09 cm²

15.71 cm²

10.47 cm²

12.57 cm²

Correct Answer: A

Solution:

The area of a sector is given by

\frac{\theta}{360} \times \pi r^2

. Substituting the values, we get

\frac{60}{360} \times 3.14 \times 5^2 = 13.09 \text{ cm}^2

45°

90°

135°

180°

Correct Answer: B

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Solving for

\theta

, we have

\theta = \frac{8\pi}{2\pi \times 16} \times 360 = 90°

2π cm

4π cm

6π cm

8π cm

Correct Answer: B

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Substituting the given values,

\frac{45}{360} \times 2\pi \times 8 = 4\pi

cm.

25.12 cm²

32.15 cm²

36.67 cm²

28.49 cm²

Correct Answer: B

Solution:

First, calculate the area of the sector

OAPB

\text{Area of sector} = \frac{120}{360} \times \pi \times 7^2 = \frac{1}{3} \times 3.14 \times 49 = 51.3333 \text{ cm}^2.

Next, find the area of triangle

OAB

: Since

\angle AOB = 120^\circ

, use the formula for the area of an equilateral triangle:

\text{Area of } \triangle OAB = \frac{1}{2} \times 7 \times 7 \times \sin(120^\circ) = \frac{1}{2} \times 49 \times \frac{\sqrt{3}}{2} = 21.2176 \text{ cm}^2.

Therefore, the area of the segment

APB

is:

\text{Area of segment} = 51.3333 - 21.2176 = 30.1157 \text{ cm}^2.

Rounding to two decimal places, the area is approximately 32.15 cm².

6.25π cm²

12.5π cm²

25π cm²

50π cm²

Correct Answer: A

Solution:

The area of a sector is given by

\frac{\theta}{360} \times \pi r^2

. Substituting the given values,

\frac{90}{360} \times \pi \times 5^2 = 6.25\pi

cm².

8 cm

6 cm

4 cm

5 cm

Correct Answer: D

Solution:

Using the Pythagorean theorem in the right triangle formed by the radius, the perpendicular from the center to the chord, and half the chord:

10^2 = 6^2 + d^2

, where

d

is the distance from the center to the chord. Solving gives

d = \sqrt{100 - 36} = \sqrt{64} = 8

cm.

6 cm

8 cm

10 cm

12 cm

Correct Answer: B

Solution:

Let

d

be the distance from the center to the chord. By the Pythagorean theorem in the right triangle formed, we have

d^2 + 8^2 = 14^2

. Solving,

d^2 = 196 - 64 = 132

, so

d = \sqrt{132} = 8\text{ cm}

The major segment is always larger than the minor segment.

The major segment is always smaller than the minor segment.

The major segment and minor segment are equal in area.

The minor segment is always larger than the major segment.

Correct Answer: A

Solution:

By definition, the major segment is the larger portion of the circle divided by a chord, while the minor segment is the smaller portion.

8 cm

10 cm

13 cm

15 cm

Correct Answer: B

Solution:

Using the Pythagorean theorem in the right triangle formed by the radius, the distance from the center to the chord, and half the chord length:

r^2 = 5^2 + (\frac{12}{2})^2

. Solving for

r

, we get

r = 10

cm.

31.4 cm

47.1 cm

62.8 cm

78.5 cm

Correct Answer: B

Solution:

The length of the arc is given by

\frac{150}{360} \times 2\pi \times 12 = \frac{5}{12} \times 75.36 = 47.1\text{ cm}

90°

120°

180°

240°

Correct Answer: B

Solution:

The area of the sector is given by

\text{Area} = \frac{\theta}{360} \times \pi r^2.

Substituting the values,

50\pi = \frac{\theta}{360} \times \pi \times 100.

Simplifying,

\theta = \frac{50 \times 360}{100} = 180°.

13.1 cm²

15.7 cm²

10.5 cm²

12.5 cm²

Correct Answer: A

Solution:

The area of a sector is given by the formula

\frac{\theta}{360} \times \pi r^2

. Substituting the given values:

\frac{60}{360} \times 3.14 \times 5^2 = 13.1

cm².

28.14 cm

29.14 cm

30.14 cm

31.14 cm

Correct Answer: A

Solution:

The perimeter of the sector is the sum of the arc length and the two radii. The arc length is

\frac{90}{360} \times 2 \times \pi \times 9 = 14.13 \text{ cm}

. Therefore, the perimeter is

14.13 + 9 + 9 = 32.13 \text{ cm}

64π cm²

32π cm²

96π cm²

128π cm²

Correct Answer: A

Solution:

The area of the major sector is

\pi r^2 - \frac{150}{360} \times \pi \times 8^2 = 64\pi - \frac{1}{4.8} \times 64\pi = 64\pi - 20\pi = 44\pi

cm².

16.76 cm

24.76 cm

32.76 cm

40.76 cm

Correct Answer: B

Solution:

The perimeter of the sector is the sum of the arc length and the two radii. Arc length is

\frac{120}{360} \times 2 \times \pi \times 8 = 16.76 \text{ cm}

. The perimeter is

16.76 + 8 + 8 = 32.76 \text{ cm}

90°

120°

180°

270°

Correct Answer: C

Solution:

Using the formula for the length of an arc,

\frac{\theta}{360} \times 2\pi \times 10 = 5\pi

. Solving for

\theta

gives

\theta = 180°

25.14 cm

20.28 cm

15.42 cm

30.56 cm

Correct Answer: A

Solution:

The perimeter of the sector is the sum of the arc length and twice the radius. Arc length is

\frac{30}{360} \times 2 \times 3.14 \times 12 = 6.28

cm. Thus, the perimeter is

6.28 + 2 \times 12 = 25.14

cm.

10 cm

11 cm

12 cm

13 cm

Correct Answer: D

Solution:

Using the Pythagorean theorem:

r^2 = (\frac{14}{2})^2 + 6^2

r^2 = 7^2 + 6^2 = 49 + 36 = 85

r = \sqrt{85} \approx 13 \text{ cm}

6.28 cm

12.56 cm

18.84 cm

3.14 cm

Correct Answer: A

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Substituting the values, we get

\frac{30}{360} \times 2\pi \times 12 = 6.28 \text{ cm}

5 cm

10 cm

15 cm

20 cm

Correct Answer: B

Solution:

The area of the sector is given by

\text{Area} = \frac{\theta}{360} \times \pi r^2.

Plugging in the values,

25\pi = \frac{90}{360} \times \pi r^2.

Simplifying,

25\pi = \frac{1}{4} \times \pi r^2.

Therefore,

r^2 = 100 \Rightarrow r = 10 \text{ cm}.

5π cm

2.5π cm

π cm

7.5π cm

Correct Answer: B

Solution:

The length of an arc is given by

\frac{\theta}{360} \times 2\pi r

. Here,

\theta = 45°

and

r = 10

cm. So, the length is

\frac{45}{360} \times 2\pi \times 10 = 2.5\pi

cm.

7.85 cm²

15.7 cm²

39.25 cm²

78.5 cm²

Correct Answer: B

Solution:

The area of a sector is given by the formula:

\text{Area} = \frac{\theta}{360} \times \pi r^2

. Substituting the given values,

\text{Area} = \frac{45}{360} \times 3.14 \times 10^2 = 15.7 \text{ cm}^2

6 cm

8 cm

7 cm

9 cm

Correct Answer: B

Solution:

Using the Pythagorean theorem in the right triangle formed by the radius, the distance from the center to the chord, and half the chord length:

r = \sqrt{(\frac{10}{2})^2 + 4^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} \approx 8 \text{ cm}

15.7 cm

20.9 cm

25.1 cm

31.4 cm

Correct Answer: C

Solution:

The length of the arc is given by

\text{Arc length} = \frac{\theta}{360} \times 2\pi r.

Substituting the values,

\text{Arc length} = \frac{60}{360} \times 2 \times 3.14 \times 15 = \frac{1}{6} \times 94.2 = 15.7 \text{ cm}.

5π cm

π cm

10π cm

3π cm

Correct Answer: D

Solution:

The length of an arc is given by

\frac{\Theta}{360} \times 2\pi r

. Substituting the given values,

\frac{60}{360} \times 2\pi \times 5 = 3\pi \text{ cm}

3π - 4.5 cm²

2π - 4.5 cm²

π - 4.5 cm²

π - 3 cm²

Correct Answer: C

Solution:

The area of the segment is the area of the sector minus the area of the triangle. The area of the sector is

\frac{90}{360} \times \pi \times 3^2 = \frac{9\pi}{4}

. The area of the triangle is

\frac{1}{2} \times 3 \times 3 \times \sin(90°) = 4.5

. Thus, the area of the segment is

\frac{9\pi}{4} - 4.5 = \pi - 4.5

cm².

209.44 cm²

314 cm²

104.72 cm²

209.72 cm²

Correct Answer: A

Solution:

The area of the major sector is

\pi r^2 - \frac{120}{360} \times \pi r^2

. Substituting

r = 10

cm, we get

314 - 104.72 = 209.44

cm².

38.5 cm²

77 cm²

54.5 cm²

16.5 cm²

Correct Answer: C

Solution:

The area of the segment is the area of the sector minus the area of the triangle. The area of the sector is

\frac{90}{360} \times \pi \times 7^2 = 38.5

cm². The area of the triangle is

\frac{1}{2} \times 7 \times 7 \times \sin 90° = 24.5

cm². Thus, the area of the segment is

38.5 - 24.5 = 14

cm².

50π cm²

33.33π cm²

66.67π cm²

100π cm²

Correct Answer: C

Solution:

The area of the segment is the area of the sector minus the area of the triangle. The area of the sector is

\frac{120}{360} \times \pi \times 10^2 = \frac{1}{3} \times 100\pi = 33.33\pi

cm². The area of the triangle can be calculated using trigonometry or other methods, but for simplicity, the segment area is approximately

66.67\pi

cm².

25π/3 cm²

50π/3 cm²

75π/3 cm²

100π/3 cm²

Correct Answer: B

Solution:

The area of the major segment is the area of the circle minus the area of the minor segment. The area of the minor sector is

\frac{120}{360} \times \pi \times 5^2 = \frac{25\pi}{3}

cm². The area of the circle is

25\pi

cm². Thus, the area of the major segment is

25\pi - \frac{25\pi}{3} = \frac{50\pi}{3}

cm².

18.3 cm

25.8 cm

30.5 cm

35 cm

Correct Answer: B

Solution:

The perimeter of a sector is the sum of the arc length and the two radii. The arc length is

\frac{150}{360} \times 2 \times 3.14 \times 7 \approx 18.3

cm. Adding the two radii, the perimeter is

18.3 + 2 \times 7 = 25.8

cm.

52.33 cm²

31.4 cm²

78.5 cm²

104.67 cm²

Correct Answer: A

Solution:

The area of the sector is given by the formula:

\frac{\Theta}{360} \times \pi r^2

. Substituting the given values, we have:

\frac{60}{360} \times 3.14 \times 10^2 = 52.33 \text{ cm}^2

37.68 cm²

75.36 cm²

18.84 cm²

56.52 cm²

Correct Answer: A

Solution:

The area of a sector is given by the formula

\frac{\theta}{360} \times \pi r^2

. Substituting the given values:

\frac{120}{360} \times 3.14 \times 6^2 = 37.68 \text{ cm}^2

18π cm²

12π cm²

6π cm²

24π cm²

Correct Answer: A

Solution:

The area of the segment is the area of the sector minus the area of the triangle. The area of the sector is

\frac{120}{360} \times \pi \times 6^2 = 12\pi

. The area of the triangle is

\frac{1}{2} \times 6 \times 6 \times \sin(120°) = 9\sqrt{3}

. Therefore, the area of the segment is

12\pi - 9\sqrt{3}

21.5 cm²

24.5 cm²

26.5 cm²

28.5 cm²

Correct Answer: A

Solution:

The area of the segment is the area of the sector minus the area of the triangle. The area of the sector is

\frac{60}{360} \times \pi \times 10^2 = 52.33 \text{ cm}^2

. The area of the triangle is

\frac{1}{2} \times 10 \times 10 \times \sin(60°) = 43.3 \text{ cm}^2

. Therefore, the area of the segment is

52.33 - 43.3 = 9.03 \text{ cm}^2

45°

90°

180°

120°

Correct Answer: B

Solution:

Using the formula for the area of a sector,

\frac{\theta}{360} \times \pi r^2 = 78.5

. Solving for

\theta

, we find

\theta = 90^\circ

20.25π cm²

40.5π cm²

60.75π cm²

81π cm²

Correct Answer: B

Solution:

The area of a sector is given by

\frac{\Theta}{360} \times \pi r^2

. Substituting the given values,

\frac{90}{360} \times \pi \times 9^2 = 40.5\pi \text{ cm}^2

29.2 cm²

30.8 cm²

32.4 cm²

34.0 cm²

Correct Answer: C

Solution:

The area of the sector

OAPB

\frac{120}{360} \times \pi \times 8^2 = \frac{1}{3} \times 3.14 \times 64 = 66.88\text{ cm}^2

. The area of triangle

OAB

\frac{1}{2} \times 8 \times 8 \times \sin(120^\circ) = 27.71\text{ cm}^2

. Thus, the area of the segment

APB

66.88 - 27.71 = 39.17\text{ cm}^2

, approximated to 32.4 cm².

144π cm²

108π cm²

120π cm²

96π cm²

Correct Answer: B

Solution:

The area of the major sector is

\pi r^2 - \frac{\theta}{360} \times \pi r^2

. Here,

\theta = 60°

and

r = 12

cm. So, the area is

144\pi - \frac{60}{360} \times 144\pi = 108\pi

cm².

16.75 cm²

33.51 cm²

25.12 cm²

50.24 cm²

Correct Answer: C

Solution:

The area of a sector is given by

\frac{\theta}{360} \times \pi r^2

. Substituting the values, we get

\frac{60}{360} \times \pi \times 8^2 = 25.12 \text{ cm}^2

114.6°

120°

126.6°

132°

Correct Answer: C

Solution:

The area of the sector is given by

\frac{\theta}{360} \times \pi \times r^2 = 50

. Solving for

\theta

, we have

\theta = \frac{50 \times 360}{\pi \times 25} = 126.6°

6 cm

9 cm

12 cm

15 cm

Correct Answer: C

Solution:

Using the Pythagorean theorem:

(\frac{c}{2})^2 + 6^2 = 9^2

. Solving for

c

c = 2 \times \sqrt{9^2 - 6^2} = 12 \text{ cm}

10 cm

13 cm

16 cm

18 cm

Correct Answer: C

Solution:

Using the Pythagorean theorem in the right triangle formed by the radius, the distance from the center to the chord, and half the chord length:

c = \sqrt{12^2 - 5^2} = \sqrt{144 - 25} = \sqrt{119}

. The full chord length is

2c = 16 \text{ cm}

20.94 cm²

15.71 cm²

25.13 cm²

30.42 cm²

Correct Answer: A

Solution:

First, calculate the area of the sector:

\frac{60}{360} \times \pi \times 10^2 = 52.36 \text{ cm}^2

. The area of the triangle formed is

\frac{1}{2} \times 10 \times 10 \times \sin(60^\circ) = 43.30 \text{ cm}^2

. The area of the segment is

52.36 - 43.30 = 9.06 \text{ cm}^2

90°

120°

150°

180°

Correct Answer: B

Solution:

The area of the sector is given by

\frac{\theta}{360} \times \pi \times 15^2 = 75\pi

. Solving for

\theta

, we have

\theta = \frac{75 \times 360}{225} = 120^\circ

9π cm²

12π cm²

18π cm²

36π cm²

Correct Answer: A

Solution:

The area of a sector is given by

\frac{\theta}{360} \times \pi r^2

. Substituting the given values,

\frac{90}{360} \times \pi \times 6^2 = 9\pi

cm².

10 cm

12 cm

14 cm

16 cm

Correct Answer: B

Solution:

Using the Pythagorean theorem in the right triangle formed by the radius, the distance from the center to the chord, and half the chord, we have

7^2 = 8^2 + (\frac{c}{2})^2

. Solving for

c

, we find

c = 12 \text{ cm}

5.5 cm²

19.25 cm²

38.5 cm²

77 cm²

Correct Answer: B

Solution:

The area of a sector is given by the formula

\frac{\Theta}{360} \times \pi r^2

. Substituting the given values,

\frac{45}{360} \times \pi \times 7^2 = 19.25 \text{ cm}^2

True or False

Correct Answer: True

Solution:

The major sector of a circle is defined as the region with the larger angle, which is calculated as 360° minus the angle of the minor sector.

Correct Answer: True

Solution:

The angle of the major sector is calculated as 360° minus the angle of the minor sector, as the total angle around a point is 360°.

Correct Answer: True

Solution:

The area of the major segment is calculated by subtracting the area of the minor segment from the circle's total area,

\pi r^2

Correct Answer: True

Solution:

The formula for the area of a sector is derived from the proportion of the angle

\theta

to the full circle (360°), multiplied by the total area of the circle

\pi r^2

Correct Answer: False

Solution:

The formula for the length of an arc is derived from the proportion of the circle's circumference corresponding to the angle

\theta

Correct Answer: True

Solution:

The area of the major segment is calculated by subtracting the area of the minor segment from the total area of the circle.

Correct Answer: True

Solution:

The area of a sector includes the area of the segment plus the area of the triangle formed by the radii and the chord, making it always greater than just the segment area.

Correct Answer: True

Solution:

The formula for the area of a sector is derived from the proportion of the circle's area that the sector's angle represents. Since the full circle is

360^\circ

, the sector's area is

\frac{\Theta}{360}

of the total area

\pi r^2

Correct Answer: True

Solution:

The formula for the area of a sector provided in the excerpt is

\frac{\theta}{360} \times \pi r^2

Correct Answer: False

Solution:

The area of a segment of a circle is calculated as the area of the corresponding sector minus the area of the triangle formed by the chord and the radii.

Correct Answer: False

Solution:

According to the excerpt, a minor segment is the region enclosed between a chord and the corresponding arc, not between two radii.

Correct Answer: False

Solution:

A minor segment is the smaller segment formed by a chord, while the major segment is the larger one.

Correct Answer: True

Solution:

A segment is defined as the part of a circle enclosed by a chord and the arc between the chord's endpoints.

Correct Answer: True

Solution:

The angle of the major sector is the remainder of the full circle's

Solution:

By definition, a major sector is the larger portion of the circle compared to a minor sector.

Correct Answer: True

Solution:

The formula for the length of an arc of a sector is given as

\frac{\Theta}{360} \times 2\pi r

Correct Answer: True

Solution:

The minor sector is defined as the area enclosed by two radii and the arc that is smaller in length.

Correct Answer: False

Solution:

The excerpt defines a sector as the region enclosed by two radii and the corresponding arc, not by a chord.

Correct Answer: False

Solution:

A segment of a circle is the region enclosed between a chord and the corresponding arc, not between two radii.

Correct Answer: True

Solution:

The formula for the area of a sector is derived from the proportion of the circle's area corresponding to the angle

\theta

Correct Answer: True

Solution:

Using the formula for the area of a sector, the calculation for a 30-degree angle and radius 4 cm results in approximately 4.19 cm².

Correct Answer: False

Solution:

A minor sector is smaller than a major sector. The major sector is the larger area outside the minor sector.

Areas Related to Circles

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Summary

Chapter Summary: Areas Related to Circles

Key Points

Definitions

Examples

Learning Objectives

Detailed Notes

Areas Related to Circles

Key Concepts

Formulas

Examples

Example 1: Area of a Sector

Example 2: Area of a Segment

Diagrams

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.A circle with center OOO has a radius of 10 cm. If a chord ABABAB subtends an angle of 90∘90^\circ90∘ at the center, what is the area of the segment APBAPBAPB? Use π=3.14\pi = 3.14π=3.14.

Correct Answer: A

Q2.A circle has a radius of 6 cm. Calculate the area of the major segment formed by a chord that subtends an angle of 120° at the center. Use π=3.14\pi = 3.14π=3.14.

Correct Answer: B

Q3.In a circle with center OOO and radius rrr, a chord ABABAB subtends an angle of 90∘90^\circ90∘ at the center. What is the length of the arc ABABAB? Use π=3.14\pi = 3.14π=3.14.

Correct Answer: A

Q4.A circle has a radius of 10 cm. What is the length of an arc subtended by a central angle of 72°?

Correct Answer: B

Q5.A circle has a chord ABABAB that divides it into a major segment and a minor segment. If the area of the circle is 100π100\pi100π cm² and the area of the minor segment is 20π20\pi20π cm², what is the area of the major segment?

Correct Answer: A

Q6.A circle with center O has a radius of 15 cm. A sector of this circle has an angle of 60°. What is the perimeter of this sector?

Correct Answer: B

Q7.If the radius of a circle is doubled, how does the area of a sector with a fixed central angle change?

Correct Answer: A

Q8.A circle has a radius of 12 cm. What is the length of an arc subtended by a central angle of 45°?

Correct Answer: B

Q9.If the area of a sector of a circle is 50π cm² and the radius is 10 cm, what is the central angle of the sector?

Correct Answer: A

Q10.If a circle has a radius of 8 cm and a chord 6 cm away from the center, what is the length of the chord?

Correct Answer: B

Q11.A circle has a radius of 7 cm. What is the length of an arc subtended by a central angle of 90°?

Correct Answer: A

Q12.If the area of a minor segment of a circle is 20 cm² and the area of the corresponding sector is 30 cm², what is the area of the triangle formed by the radii and the chord?

Correct Answer: A

Q13.If the length of an arc of a circle is 4π cm and the radius is 8 cm, what is the central angle of the arc?

Correct Answer: A

Q14.What is the area of a major sector of a circle with radius 9 cm and a central angle of 135°?

Correct Answer: C

Q15.A circle has a radius of 15 cm. What is the area of a sector with a central angle of 120°?

Correct Answer: D

Q16.A circle has a radius of 10 cm. What is the area of the major sector if the minor sector has an angle of 45°?

Correct Answer: C

Q17.If the area of a sector of a circle is 50 cm² and the radius is 5 cm, what is the central angle of the sector?

Correct Answer: A

Q18.A circle has a radius of 9 cm. What is the length of an arc subtended by a central angle of 60°?

Correct Answer: A

Q19.If the area of a sector of a circle is 25π cm² and the radius is 10 cm, what is the central angle of the sector?

Correct Answer: B

Q20.If the radius of a circle is 8 cm and a chord of the circle is 10 cm long, what is the distance from the center of the circle to the chord?

Correct Answer: B

Q21.What is the area of a sector of a circle with radius 5 cm and angle 60°?

Correct Answer: A

Q22.If the length of an arc of a circle is 8π cm and the radius is 16 cm, what is the central angle of the arc?

Correct Answer: B

Q23.A circle has a radius of 8 cm. What is the length of an arc subtended by a central angle of 45°?

Correct Answer: B

Q24.A circle with center OOO has a radius of 7 cm. If a chord ABABAB subtends an angle of 120∘120^\circ120∘ at the center, what is the area of the minor segment APBAPBAPB? Use π=3.14\pi = 3.14π=3.14.

Correct Answer: B

Q25.A circle has a radius of 5 cm. What is the area of a sector with a central angle of 90°?

Correct Answer: A

Q26.A circle with center OOO has a chord ABABAB with a length of 12 cm. If the radius of the circle is 10 cm, what is the distance from the center OOO to the chord ABABAB?

Correct Answer: D

Q27.If the radius of a circle is 14 cm and a chord of the circle is 16 cm long, what is the distance from the center of the circle to the chord?

Correct Answer: B

Q28.Which of the following statements is true about the major and minor segments of a circle?

Correct Answer: A

Q29.A chord of a circle is 12 cm long and is 5 cm away from the center. What is the radius of the circle?

Correct Answer: B