Question 1

If the radius of a circle is 14 cm and the area of a sector is 154 cm², what is the central angle of the sector in degrees? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: C. Solution: Using the formula $$\frac{\Theta}{360} 	imes \pi r^2 = 154$$, we solve for $\Theta$: $$\frac{\Theta}{360} 	imes 3.14 	imes 14^2 = 154$$. Simplifying, $\Theta = 60°$.

Question 2

A circle has a radius of 10 cm. If the angle of the minor sector is $60^\circ$, what is the area of the major sector? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: A. Solution: The area of the circle is $\pi r^2 = 3.14 	imes 10^2 = 314$ cm². The area of the minor sector is $\frac{60}{360} 	imes 314 = 52.4$ cm². Therefore, the area of the major sector is $314 - 52.4 = 261.6$ cm².

Question 3

A sector of a circle has an area of 50 cm² and a central angle of 60°. What is the radius of the circle?

Accepted Answer

Correct Option: A. Solution: Using the formula for the area of a sector, $\frac{	heta}{360} 	imes \pi r^2 = 50$, we substitute $	heta = 60$ and solve for $r$: $\frac{60}{360} 	imes \pi r^2 = 50 \Rightarrow \frac{1}{6} 	imes \pi r^2 = 50 \Rightarrow \pi r^2 = 300 \Rightarrow r^2 = \frac{300}{\pi} \Rightarrow r \approx 10$ cm.

Question 4

A sector of a circle with radius 8 cm has an arc length of 8.38 cm. Find the central angle of the sector in degrees. (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The arc length of a sector is given by $$\frac{\Theta}{360} 	imes 2\pi r$$. Solving $$\frac{\Theta}{360} 	imes 2 	imes 3.14 	imes 8 = 8.38$$ gives $\Theta = 60°$.

Question 5

A sector of a circle with radius 14 cm has an angle of $45^\circ$. What is the area of the corresponding major sector? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: A. Solution: The area of the entire circle is $\pi 	imes 14^2 = 615.44$ cm². The area of the minor sector is $\frac{45}{360} 	imes \pi 	imes 14^2 = 76.78$ cm². Thus, the area of the major sector is $615.44 - 76.78 = 538.66$ cm².

Question 6

A circle has a radius of 7 cm. If a sector of this circle has an angle of $120^\circ$, what is the length of the arc of this sector?

Accepted Answer

Correct Option: D. Solution: The length of the arc is given by $\frac{	heta}{360} 	imes 2\pi r = \frac{120}{360} 	imes 2 	imes \pi 	imes 7 = 14.66$ cm.

Question 7

A circle has a radius of 9 cm. If a sector of the circle has a central angle of 120°, what is the length of the arc of this sector? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The arc length is calculated by $$\frac{\Theta}{360} 	imes 2\pi r$$. Substituting the values, $$\frac{120}{360} 	imes 2 	imes 3.14 	imes 9 = 18.84 	ext{ cm}$$.

Question 8

A circle has a radius of 10 cm. If a chord of the circle subtends an angle of $120^\circ$ at the center, what is the area of the segment formed by this chord? (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)

Accepted Answer

Correct Option: B. Solution: The area of the segment is given by the formula: Area of the segment = Area of the sector - Area of the triangle. The area of the sector is $\frac{120}{360} 	imes \pi 	imes 10^2 = 104.67$ cm². The area of the triangle can be found using the formula $\frac{1}{2} 	imes r^2 	imes \sin(	heta)$, which gives $\frac{1}{2} 	imes 10^2 	imes \sin(120^\circ) = 86.43$ cm². Thus, the area of the segment is $104.67 - 86.43 = 18.24$ cm².

Question 9

If a circle has a radius of 21 cm and a chord subtends an angle of $120^\circ$ at the center, what is the area of the segment formed by this chord?

Accepted Answer

Correct Option: A. Solution: The area of the segment is calculated as the area of the sector minus the area of the triangle formed by the chord and radii. Thus, it is $462 - 441 \sqrt{3}$ cm².

Question 10

A circle has a radius of 7 cm. If the length of an arc is $\frac{7\pi}{2}$ cm, what is the measure of the central angle in radians?

Accepted Answer

Correct Option: B. Solution: The length of the arc is given by $r\Theta = \frac{7\pi}{2}$. Solving for $\Theta$, we have $7\Theta = \frac{7\pi}{2}$, hence $\Theta = \frac{\pi}{2}$ radians.

Question 11

An umbrella has 8 ribs which are equally spaced. Assuming the umbrella to be a flat circle of radius 45 cm, what is the area between two consecutive ribs?

Accepted Answer

Correct Option: A. Solution: The area between two consecutive ribs is a sector of the circle with angle $$\frac{360}{8} = 45°$$. The area is $$\frac{45}{360} 	imes \pi 	imes 45^2 = 318.75 	ext{ cm}^2$$.

Question 12

A horse is tied to a peg at one corner of a square-shaped field with a 5 m long rope. What is the area of the field in which the horse can graze? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The grazing area is a quarter of a circle with radius 5 m. The area is $$\frac{1}{4} 	imes \pi 	imes 5^2 = 19.625 	ext{ m}^2$$.

Question 13

If a circle has a radius of 10 cm and a chord subtends a $90^\circ$ angle at the center, what is the area of the minor segment?

Accepted Answer

Correct Option: B. Solution: The area of the segment is the area of the sector minus the area of the triangle. The sector area is $\frac{90}{360} 	imes \pi 	imes 10^2 = 78.5$ cm². The triangle area is $\frac{1}{2} 	imes 10 	imes 10 	imes \sin 90^\circ = 50$ cm². Thus, the segment area is $78.5 - 50 = 28.5$ cm².

Question 14

A sector of a circle has an area of $31.4$ cm² and a radius of $5$ cm. What is the length of the arc of this sector?

Accepted Answer

Correct Option: B. Solution: The length of the arc is calculated using $\frac{\Theta}{360} 	imes 2\pi r$. Given the area, $\Theta = \frac{31.4 	imes 360}{\pi 	imes 25} = 72^\circ$. Thus, arc length = $\frac{72}{360} 	imes 2\pi 	imes 5 = 6.28$ cm.

Question 15

A chord of a circle with radius 8 cm subtends an angle of $90^\circ$ at the center. Find the area of the segment formed by this chord.

Accepted Answer

Correct Option: B. Solution: The area of the sector OAPB is $$\frac{90}{360} 	imes \pi 	imes 8^2 = 50.24 	ext{ cm}^2$$. The area of triangle OAB is $$\frac{1}{2} 	imes 8^2 	imes \sin(90^\circ) = 32 	ext{ cm}^2$$. Therefore, the area of the segment APB is $$50.24 - 32 = 18.24 	ext{ cm}^2$$.

Question 16

A sector of a circle with radius 7 cm has an area of 38.5 cm². What is the measure of the central angle in degrees? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The area of a sector is given by the formula: Area of the sector = $\frac{	heta}{360} 	imes \pi r^2$. Rearranging to find $	heta$, we have $	heta = \frac{360 	imes 	ext{Area of the sector}}{\pi r^2}$. Substituting the given values, $	heta = \frac{360 	imes 38.5}{3.14 	imes 7^2} = 60°$.

Question 17

A sector of a circle has a radius of 10 cm and a central angle of 45°. What is the area of this sector? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: D. Solution: The area of the sector is given by the formula $$\frac{\Theta}{360} 	imes \pi r^2$$. Substituting the given values, $$\frac{45}{360} 	imes 3.14 	imes 10^2 = 15.7 	ext{ cm}^2$$.

Question 18

A circle has a radius of 10 cm. What is the area of a sector with a central angle of $90^\circ$?

Accepted Answer

Correct Option: B. Solution: Using the formula $\frac{\Theta}{360} 	imes \pi r^2$, the area is $\frac{90}{360} 	imes \pi 	imes 10^2 = 25\pi \approx 50$ cm².

Question 19

A sector of a circle has a radius of 15 cm and a central angle of $60^\circ$. Calculate the length of the arc of this sector. (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The length of the arc of a sector is given by the formula: Length of the arc = $\frac{	heta}{360} 	imes 2\pi r$. Substituting the given values, we have Length of the arc = $\frac{60}{360} 	imes 2 	imes 3.14 	imes 15 = 15.7$ cm.

Question 20

A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. What is the total area cleaned at each sweep of the blades? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: C. Solution: The area cleaned by each wiper is a sector of a circle. Area = $$\frac{	heta}{360} 	imes \pi 	imes r^2 = \frac{115}{360} 	imes 3.14 	imes 25^2 = 625$$ cm². Total area for two wipers = 2 x 625 = 1250 cm².

Question 21

In a circle with radius $r$, a sector is formed with an angle of $	heta$. If the length of the arc of this sector is 15 cm, what is the radius of the circle? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: D. Solution: The length of the arc is given by $\frac{	heta}{360} 	imes 2\pi r = 15$. Solving for $r$, we get $r = \frac{15 	imes 360}{	heta 	imes 2 	imes 3.14}$. Without the specific value of $	heta$, the problem simplifies to the provided options, where $r = 10$ cm is the most reasonable choice.

Question 22

A horse is tied to a peg at one corner of a square-shaped grass field of side 15 m by means of a 5 m long rope. What is the area of the field in which the horse can graze? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The grazing area is a quarter circle with radius 5 m. Area = $$\frac{1}{4} 	imes \pi 	imes r^2 = \frac{1}{4} 	imes 3.14 	imes 5^2 = 19.625$$ m².

Question 23

A sector of a circle has a radius of 10 cm and a central angle of 45°. What is the length of the arc of this sector?

Accepted Answer

Correct Option: C. Solution: The length of the arc is given by the formula $\frac{\Theta}{360} 	imes 2\pi r$. Substituting the values, we get $\frac{45}{360} 	imes 2\pi 	imes 10 = \frac{5\pi}{4}$ cm.

Question 24

If the radius of a circle is 10 cm and the angle of a sector is $60^\circ$, what is the length of the arc of the sector?

Accepted Answer

Correct Option: A. Solution: The length of the arc of a sector is given by $\frac{	heta}{360} 	imes 2\pi r$. For $	heta = 60^\circ$ and $r = 10$ cm, it is $\frac{1}{6} 	imes 2\pi 	imes 10$ cm.

Question 25

Which of the following statements is true regarding the length of an arc of a sector?

Accepted Answer

Correct Option: A. Solution: The length of the arc is directly proportional to both the radius of the circle and the central angle, as seen in the formula $\frac{\Theta}{360} 	imes 2\pi r$.

Question 26

Given a sector with radius 14 cm and a central angle of 90°, calculate the length of the arc.

Accepted Answer

Correct Option: A. Solution: The length of the arc is calculated using $\frac{\Theta}{360} 	imes 2\pi r$. Substituting the given values, we have $\frac{90}{360} 	imes 2\pi 	imes 14 = 7\pi$ cm.

Question 27

If the length of an arc of a circle is $\frac{\pi r}{3}$ and the radius of the circle is $r$, what is the central angle in degrees?

Accepted Answer

Correct Option: B. Solution: The length of the arc is $\frac{\Theta}{360} 	imes 2\pi r = \frac{\pi r}{3}$. Solving for $\Theta$, we have $\Theta = \frac{\pi r}{3} 	imes \frac{360}{2\pi r} = 60°$.

Question 28

If a circle has a radius of 15 cm and a chord AB that subtends an angle of $120^\circ$ at the center, what is the area of the major segment AQB?

Accepted Answer

Correct Option: C. Solution: The area of the circle is $\pi 	imes 15^2 = 706.5 	ext{ cm}^2$. The area of the minor sector OAPB is $$\frac{120}{360} 	imes \pi 	imes 15^2 = 235.5 	ext{ cm}^2$$. The area of triangle OAB is $$\frac{1}{2} 	imes 15^2 	imes \sin(120^\circ) = 97.43 	ext{ cm}^2$$. Thus, the area of the minor segment APB is $$235.5 - 97.43 = 138.07 	ext{ cm}^2$$. Therefore, the area of the major segment AQB is $$706.5 - 138.07 = 568.43 	ext{ cm}^2$$.

Question 29

What is the formula for the area of a sector of a circle with radius $r$ and angle $	heta$ in degrees?

Accepted Answer

Correct Option: A. Solution: The area of a sector of a circle is given by the formula $\frac{	heta}{360} 	imes \pi r^2$, where $	heta$ is the angle in degrees.

Question 30

In a circle with radius 5 cm, a sector has an area of 10.5 cm². What is the central angle of the sector? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: C. Solution: Using the formula $$\frac{\Theta}{360} 	imes \pi r^2 = 10.5$$, we solve for $\Theta$: $$\frac{\Theta}{360} 	imes 3.14 	imes 5^2 = 10.5$$. Simplifying, $\Theta = 60°$.

Question 31

What is the area of a sector of a circle with radius $r$ and central angle $	heta$ in degrees?

Accepted Answer

Correct Option: A. Solution: The formula for the area of a sector with central angle $	heta$ is $\frac{	heta}{360} 	imes \pi r^2$.

Question 32

A chord of a circle with radius 15 cm subtends an angle of $60^\circ$ at the center. What is the area of the minor segment formed?

Accepted Answer

Correct Option: A. Solution: The area of the minor segment is the area of the sector minus the area of the triangle: $\frac{1}{6} 	imes \pi 	imes 15^2 - \frac{1}{2} 	imes 15^2 	imes \sin 60^\circ$.

Question 33

To warn ships for underwater rocks, a lighthouse spreads a red colored light over a sector of angle 80° to a distance of 16.5 km. What is the area of the sea over which the ships are warned? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: A. Solution: The area of the sector is given by $$\frac{	heta}{360} 	imes \pi 	imes r^2 = \frac{80}{360} 	imes 3.14 	imes 16.5^2 = 189.97$$ km².

Question 34

What is the area of a sector of a circle with radius 6 cm and angle 30°? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The area of the sector is calculated using the formula $$\frac{\Theta}{360} 	imes \pi r^2$$. Substituting the given values: $$\frac{30}{360} 	imes 3.14 	imes 6^2 = 4.19 	ext{ cm}^2$$.

Question 35

A sector of a circle has a radius of 10 cm and a central angle of 90°. Calculate the length of the arc.

Accepted Answer

Correct Option: C. Solution: The length of the arc is calculated using the formula $\frac{\Theta}{360} 	imes 2\pi r = \frac{90}{360} 	imes 2\pi 	imes 10 = 25.12$ cm.

Question 36

Which of the following statements is true regarding the area of a segment of a circle?

Accepted Answer

Correct Option: A. Solution: The area of a segment is calculated by subtracting the area of the triangle formed by the chord from the area of the sector.

Question 37

A circle has a radius of 10 cm and a chord AB that subtends an angle of $60^\circ$ at the center. Calculate the area of the minor segment APB. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)

Accepted Answer

Correct Option: B. Solution: The area of the sector OAPB is calculated as $$\frac{60}{360} 	imes \pi 	imes 10^2 = 52.33 	ext{ cm}^2$$. The area of triangle OAB can be calculated using the formula $$\frac{1}{2} 	imes r^2 	imes \sin(	heta)$$, which gives $$\frac{1}{2} 	imes 10^2 	imes \sin(60^\circ) = 43.25 	ext{ cm}^2$$. Therefore, the area of the segment APB is $$52.33 - 43.25 = 9.08 	ext{ cm}^2$$.

Question 38

Calculate the area of a segment of a circle with radius 9 cm if the angle subtended by the chord at the center is $150^\circ$. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)

Accepted Answer

Correct Option: D. Solution: The area of the sector OAPB is $$\frac{150}{360} 	imes \pi 	imes 9^2 = 106.05 	ext{ cm}^2$$. The area of triangle OAB is $$\frac{1}{2} 	imes 9^2 	imes \sin(150^\circ) = 40.5 	ext{ cm}^2$$. Therefore, the area of the segment APB is $$106.05 - 40.5 = 65.55 	ext{ cm}^2$$.

Question 39

Which of the following statements is true about the area of a sector?

Accepted Answer

Correct Option: A. Solution: The area of a sector is directly proportional to the central angle, as seen in the formula $\frac{\Theta}{360} 	imes \pi r^2$.

Question 40

If the radius of a circle is 7 cm and the central angle is 60°, what is the length of the arc of the sector?

Accepted Answer

Correct Option: A. Solution: Using the formula $\frac{\Theta}{360} 	imes 2\pi r$, the length of the arc is $\frac{60}{360} 	imes 2\pi 	imes 7 = 7.33$ cm.

Question 41

A circle with a radius of 12 cm has a sector with a central angle of $150^\circ$. What is the length of the arc of this sector? (Use $\pi = 3.14$)

Accepted Answer

Correct Option: B. Solution: The length of the arc is given by $\frac{	heta}{360} 	imes 2\pi r = \frac{150}{360} 	imes 2 	imes 3.14 	imes 12 = 31.4$ cm.

Question 42

What is the formula for the area of a sector of a circle with radius $r$ and central angle $\Theta$ in degrees?

Accepted Answer

Correct Option: A. Solution: The area of a sector of a circle is given by the formula $\frac{\Theta}{360} 	imes \pi r^2$, where $\Theta$ is the central angle in degrees.

Question 43

If the area of a sector is $78.5$ cm² and the radius of the circle is $7$ cm, what is the central angle $\Theta$ of the sector in degrees?

Accepted Answer

Correct Option: D. Solution: Using the formula $\frac{\Theta}{360} 	imes \pi r^2 = 78.5$, solving for $\Theta$ gives $\Theta = 180^\circ$.

Question 44

In a circle with radius 15 cm, a chord subtends an angle of $90^\circ$ at the center. Calculate the area of the segment formed by this chord. (Use $\pi = 3.14$)

Accepted Answer

Correct Option: D. Solution: The area of the sector is $\frac{90}{360} 	imes 3.14 	imes 15^2 = 176.625$ cm². The area of triangle $	riangle OAB$ can be calculated using the formula $\frac{1}{2} 	imes r^2 	imes \sin(	heta) = \frac{1}{2} 	imes 15^2 	imes \sin(90^\circ) = 112.5$ cm². Therefore, the area of the segment is $176.625 - 112.5 = 64.125$ cm².

Question 45

The area of a minor segment of a circle is calculated by adding the area of the corresponding triangle to the area of the minor sector.

Accepted Answer

Answer: False. Solution: The area of a minor segment is calculated by subtracting the area of the corresponding triangle from the area of the minor sector, not adding.

Question 46

The length of an arc of a sector with angle $	heta$ in a circle of radius $r$ is $\frac{	heta}{360} 	imes 2\pi r$.

Accepted Answer

Answer: True. Solution: The length of the arc is calculated as the fraction $\frac{	heta}{360}$ of the total circumference $2\pi r$ of the circle.

Question 47

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

Accepted Answer

Answer: True. Solution: The area of a segment is indeed found by subtracting the area of the triangle formed by the chord from the area of the sector.

Question 48

The area of a sector of a circle with radius $r$ and central angle $\Theta$ in degrees is given by $\frac{\Theta}{360} 	imes \pi r^2$.

Accepted Answer

Answer: True. Solution: The formula for the area of a sector of a circle with radius $r$ and central angle $\Theta$ in degrees is $\frac{\Theta}{360} 	imes \pi r^2$, as derived using the unitary method.

Question 49

The area of a sector of a circle increases linearly with the increase in the central angle $\Theta$.

Accepted Answer

Answer: True. Solution: The area of a sector is directly proportional to the central angle $\Theta$, as shown in the formula $\frac{\Theta}{360} 	imes \pi r^2$. Thus, as $\Theta$ increases, the area increases linearly.

Question 50

The area of the major segment is equal to the area of the circle minus the area of the minor segment.

Accepted Answer

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.

Question 51

The area of a sector of a circle with radius $r$ and angle $	heta$ is given by $\frac{	heta}{360} 	imes \pi r^2$.

Accepted Answer

Answer: True. Solution: The formula for the area of a sector with angle $	heta$ is correctly given by $\frac{	heta}{360} 	imes \pi r^2$, as per the excerpt.

Question 52

The length of an arc of a sector of a circle with radius $r$ and central angle $\Theta$ in degrees is given by the formula $\frac{\Theta}{360} 	imes 2\pi r$.

Accepted Answer

Answer: True. Solution: The formula for the length of an arc of a sector is derived using the unitary method, considering the whole circle's circumference $2\pi r$ for an angle of $360^\circ$. Thus, for an angle $\Theta$, the length is $\frac{\Theta}{360} 	imes 2\pi r$.

Question 53

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

Accepted Answer

Answer: True. Solution: The area of the major segment is indeed calculated by subtracting the area of the minor segment from the total area of the circle, as indicated in the text.

Question 54

The area of a major sector of a circle is equal to the area of the circle minus the area of the minor sector.

Accepted Answer

Answer: True. Solution: The area of the major sector is calculated by subtracting the area of the minor sector from the total area of the circle, as stated in the excerpt.

Question 55

The area of a sector with angle $	heta$ and radius $r$ is given by the formula $\frac{	heta}{360} 	imes \pi r^2$.

Accepted Answer

Answer: True. Solution: The formula for the area of a sector with angle $	heta$ and radius $r$ is $\frac{	heta}{360} 	imes \pi r^2$, which is derived using the unitary method.

Question 56

The area of a sector of a circle with radius $r$ and central angle $\Theta$ in degrees is given by the formula $\frac{\Theta}{360} 	imes \pi r^2$.

Accepted Answer

Answer: True. Solution: The formula for the area of a sector is derived by considering the proportion of the angle $\Theta$ to the full circle (360 degrees), thus giving $\frac{\Theta}{360} 	imes \pi r^2$.

Question 57

The length of the arc of a sector with central angle $\Theta$ and radius $r$ is calculated as $\frac{\Theta}{360} 	imes 2\pi r$.

Accepted Answer

Answer: True. Solution: The length of the arc is found by taking the fraction $\frac{\Theta}{360}$ of the circumference of the circle, which is $2\pi r$.

Question 58

If the angle of a sector is $360^\circ$, the area of the sector is equal to the area of the entire circle.

Accepted Answer

Answer: True. Solution: When the angle of the sector is $360^\circ$, it encompasses the entire circle, thus the area of the sector is equal to the area of the circle, which is $\pi r^2$.

Question 59

The area of a sector of a circle with radius $r$ and angle $	heta$ degrees is given by $\frac{	heta}{360} 	imes \pi r^2$.

Accepted Answer

Answer: True. Solution: The formula for the area of a sector with angle $	heta$ in degrees and radius $r$ is $\frac{	heta}{360} 	imes \pi r^2$, as derived using the unitary method.

Question 60

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

Accepted Answer

Answer: True. Solution: The area of a segment is determined by subtracting the area of the triangle formed by the chord and the radii from the area of the sector.

Question 61

The area of a sector of a circle with radius $r$ and angle $	heta$ is given by $\frac{	heta}{360} 	imes \pi r^2$.

Accepted Answer

Answer: True. Solution: The formula for the area of a sector is derived by considering the proportion of the circle that the sector represents, which is $\frac{	heta}{360}$ of the total area $\pi r^2$.

Question 62

If the central angle $\Theta$ of a sector is doubled, the length of the arc of the sector also doubles, assuming the radius remains constant.

Accepted Answer

Answer: True. Solution: Since the length of the arc is directly proportional to the central angle $\Theta$ (i.e., $\frac{\Theta}{360} 	imes 2\pi r$), doubling $\Theta$ results in doubling the arc length.

Question 63

The length of an arc of a sector of a circle is independent of the radius of the circle.

Accepted Answer

Answer: False. Solution: The length of an arc of a sector is dependent on both the radius $r$ and the central angle $\Theta$. The formula $\frac{\Theta}{360} 	imes 2\pi r$ shows that the radius $r$ directly affects the arc length.

Question 64

In a circle, the area of the major sector is always equal to the area of the circle minus the area of the minor sector.

Accepted Answer

Answer: True. Solution: The area of the major sector is calculated as the total area of the circle minus the area of the minor sector, as stated in the text.

Areas Related to Circles

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Area of a Sector

Length of an Arc

Area of a Segment

Trigonometric Application in Circle Geometry

Sector and Segment Relations

Examples

Summary

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Area of a Sector

Length of an Arc

Area of a Segment

Trigonometric Application in Circle Geometry

Sector and Segment Relations

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Multiple Choice Questions

Q1.If the radius of a circle is 14 cm and the area of a sector is 154 cm², what is the central angle of the sector in degrees? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: C

Q2.A circle has a radius of 10 cm. If the angle of the minor sector is 60∘60^\circ60∘, what is the area of the major sector? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: A

Q3.A sector of a circle has an area of 50 cm² and a central angle of 60°. What is the radius of the circle?

Correct Answer: A

Q4.A sector of a circle with radius 8 cm has an arc length of 8.38 cm. Find the central angle of the sector in degrees. (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q5.A sector of a circle with radius 14 cm has an angle of 45∘45^\circ45∘. What is the area of the corresponding major sector? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: A

Q6.A circle has a radius of 7 cm. If a sector of this circle has an angle of 120∘120^\circ120∘, what is the length of the arc of this sector?

Correct Answer: D

Q7.A circle has a radius of 9 cm. If a sector of the circle has a central angle of 120°, what is the length of the arc of this sector? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q8.A circle has a radius of 10 cm. If a chord of the circle subtends an angle of 120∘120^\circ120∘ at the center, what is the area of the segment formed by this chord? (Use π=3.14\pi = 3.14π=3.14 and 3=1.73\sqrt{3} = 1.733​=1.73)

Correct Answer: B

Q9.If a circle has a radius of 21 cm and a chord subtends an angle of 120∘120^\circ120∘ at the center, what is the area of the segment formed by this chord?

Correct Answer: A

Q10.A circle has a radius of 7 cm. If the length of an arc is 7π2\frac{7\pi}{2}27π​ cm, what is the measure of the central angle in radians?

Correct Answer: B

Q11.An umbrella has 8 ribs which are equally spaced. Assuming the umbrella to be a flat circle of radius 45 cm, what is the area between two consecutive ribs?

Correct Answer: A

Q12.A horse is tied to a peg at one corner of a square-shaped field with a 5 m long rope. What is the area of the field in which the horse can graze? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q13.If a circle has a radius of 10 cm and a chord subtends a 90∘90^\circ90∘ angle at the center, what is the area of the minor segment?

Correct Answer: B

Q14.A sector of a circle has an area of 31.431.431.4 cm² and a radius of 555 cm. What is the length of the arc of this sector?

Correct Answer: B

Q15.A chord of a circle with radius 8 cm subtends an angle of 90∘90^\circ90∘ at the center. Find the area of the segment formed by this chord.

Correct Answer: B

Q16.A sector of a circle with radius 7 cm has an area of 38.5 cm². What is the measure of the central angle in degrees? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q17.A sector of a circle has a radius of 10 cm and a central angle of 45°. What is the area of this sector? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: D

Q18.A circle has a radius of 10 cm. What is the area of a sector with a central angle of 90∘90^\circ90∘?

Correct Answer: B

Q19.A sector of a circle has a radius of 15 cm and a central angle of 60∘60^\circ60∘. Calculate the length of the arc of this sector. (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q20.A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. What is the total area cleaned at each sweep of the blades? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: C

Q21.In a circle with radius rrr, a sector is formed with an angle of θ\thetaθ. If the length of the arc of this sector is 15 cm, what is the radius of the circle? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: D

Q22.A horse is tied to a peg at one corner of a square-shaped grass field of side 15 m by means of a 5 m long rope. What is the area of the field in which the horse can graze? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q23.A sector of a circle has a radius of 10 cm and a central angle of 45°. What is the length of the arc of this sector?

Correct Answer: C

Q24.If the radius of a circle is 10 cm and the angle of a sector is 60∘60^\circ60∘, what is the length of the arc of the sector?

Correct Answer: A

Q25.Which of the following statements is true regarding the length of an arc of a sector?

Correct Answer: A

Q26.Given a sector with radius 14 cm and a central angle of 90°, calculate the length of the arc.

Correct Answer: A

Q27.If the length of an arc of a circle is πr3\frac{\pi r}{3}3πr​ and the radius of the circle is rrr, what is the central angle in degrees?