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

Objectives Notes Exam Tips Practice Qs Ask AI

Learning Objectives

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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.

Revision Notes & Summary

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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

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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.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

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}

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

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.

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

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}

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°.

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

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}

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}

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.

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}.

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°

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

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}

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.

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.

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

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}.

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.

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.

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°

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

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.

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.

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

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.

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.

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

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.

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.

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.

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.

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}

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.

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

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°

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°

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.

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

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°

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

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

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}

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.

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}

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.

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}

6.25π 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.

\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}

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}

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.

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.

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

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.

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°

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

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 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}

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

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}

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}

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}

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

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.

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.

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.

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.

True or False

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: False

Solution:

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

Correct Answer: True

Solution:

The area of a segment is calculated by subtracting the area of the triangle formed by the chord from the area of the corresponding sector.

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: 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 calculated as 360° minus the angle of the minor sector.

Correct Answer: True

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:

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: 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 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 excerpt states that the area of the major sector is equal to the total area of the circle minus the area of the minor sector.

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: 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:

By definition, the major sector of a circle is the larger area enclosed by two radii and the arc between them.

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 the remainder of the full circle's

360^\circ

after subtracting the angle of the minor sector.

Correct Answer: True

Solution:

A sector is defined as the portion of a circle enclosed by two radii and the arc between them.

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: 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:

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

Correct Answer: False

Solution:

The area of a segment is the area of the corresponding sector minus the area of the triangle formed by the chord.

Correct Answer: True

Solution:

The area of a segment is calculated by subtracting the area of the triangle from the area of the sector.

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:

The area of a segment of a circle is actually the area of the corresponding sector minus the area of the corresponding triangle.

Correct Answer: True

Solution:

A segment of a circle is the region enclosed between a chord and the arc that it subtends.

Correct Answer: False

Solution:

The angle of the major sector is calculated as 360° minus the angle of the minor sector.

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: 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 major segment is determined by the larger area, not necessarily by containing the center of the circle.

Correct Answer: True

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: False

Solution:

The angle of a major sector is greater than 180 degrees as it is the larger portion of the circle.

Correct Answer: True

Solution:

The formula for the length of an arc is derived from the proportion of the angle

\theta

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

2\pi r

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: True

Solution:

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

Correct Answer: False

Solution:

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

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°.

Areas Related to Circles

Learning Objectives

Revision Notes & Summary

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

AI Learning Assistant

Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.A circle with center O has a radius of 9 cm. A sector of this circle has an angle of 90°. What is the perimeter of this sector?

Correct Answer: A

Q2.A circle has a radius of 9 cm. What is the area of a sector with a central angle of 90°?

Correct Answer: B

Q3.A circle has a radius of 7 cm. What is the area of the minor segment if the central angle is 90°?

Correct Answer: C

Q4.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

Q5.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

Q6.A sector of a circle has an area of 50π50\pi50π cm² and a radius of 10 cm. What is the angle of the sector in degrees?

Correct Answer: B

Q7.If the area of a sector of a circle is 78.5 cm² and the radius is 10 cm, what is the central angle of the sector?

Correct Answer: B

Q8.In a circle with radius 9 cm, a chord is 6 cm away from the center. What is the length of the chord?

Correct Answer: C

Q9.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

Q10.What is the length of an arc of a circle with radius 10 cm and a central angle of 45°?

Correct Answer: B

Q11.In a circle with radius rrr, a sector has an angle of θ\thetaθ degrees. If the area of the sector is 25π25\pi25π cm², what is the value of rrr when θ=90∘\theta = 90^\circθ=90∘?

Correct Answer: B

Q12.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

Q13.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

Q14.In a circle with radius 7 cm, a chord is 8 cm away from the center. What is the length of the chord?

Correct Answer: B

Q15.What is the area of the major segment of a circle with radius 5 cm and a central angle of 120°?

Correct Answer: B

Q16.If the length of an arc of a circle is 5π cm and the radius is 10 cm, what is the central angle of the arc?

Correct Answer: C

Q17.A circle with center O has a radius of 12 cm. A sector of this circle has an angle of 30°. What is the perimeter of this sector?

Correct Answer: A

Q18.A sector of a circle has an area of 75π75\pi75π cm² and a radius of 15 cm. What is the angle of the sector in degrees?

Correct Answer: B

Q19.A circle with center OOO and radius 15 cm has a chord ABABAB that subtends an angle of 60∘60^\circ60∘ at the center. What is the length of the arc ABABAB?

Correct Answer: C

Q20.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

Q21.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

Q22.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

Q23.A circle has a radius of 10 cm. If a chord subtends an angle of 60° at the center, what is the area of the segment formed by this chord? Use π=3.14\pi = 3.14π=3.14.

Correct Answer: A

Q24.What is the area of a major sector of a circle with radius 10 cm and a central angle of 120°?

Correct Answer: A

Q25.If the radius of a circle is doubled, how does the area of a sector with a fixed central angle change?

Correct Answer: A

Q26.A circle has a radius of 15 cm. What is the area of a sector with a central angle of 120°?

Correct Answer: D

Q27.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

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

Correct Answer: A

Q29.A circle has a radius of 6 cm. If the angle of a sector is 120°, what is the area of the sector? Use π=3.14\pi = 3.14π=3.14.

Correct Answer: A

Q30.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

Q31.A circle with center O has a radius of 7 cm. A sector of this circle has an angle of 150°. What is the perimeter of this sector?

Correct Answer: B

Q32.What is the area of a major sector of a circle with radius 8 cm and a central angle of 150°?