RBSE Solutions Class 9 Maths Chapter 13 Angles and their Measurement Important Questions

Multiple Choice Questions

 

Question 1. 120° in radians is equal to:
(a) 3\( \pi \)
(b) \( \frac {3\pi }{ 2 } \)
(c) \( \frac {2\pi }{3} \)
(d) \( \frac {\pi }{2} \)
Answer: (c) \( \frac {2\pi }{3} \)
In simple words: To change degrees to radians, you multiply the degree value by \( \frac{\pi}{180^\circ} \). For 120 degrees, this calculation gives \( \frac{2\pi}{3} \) radians. This method helps us convert angle measurements between two common systems.

🎯 Exam Tip: Remember the conversion factor: \( 1^\circ = \frac{\pi}{180} \) radians. This is a fundamental concept for angle conversions.

 

Question 2. In the given figure, O is the centre of the circle. Then (\( \angle\text{BAD} + \angle\text{BCD} \)) is equal to:

(a) 2\( \pi \) radian
(b) \( \frac {\pi }{ 2 } \) radian
(c) \( \frac {\pi }{4} \) radian
(d) \( \pi \) radian
Answer: (d) \( \pi \) radian
In simple words: In a circle, if a quadrilateral is drawn with all its corners touching the circle's edge, it's called a cyclic quadrilateral. For any cyclic quadrilateral, the opposite angles always add up to 180 degrees. Since 180 degrees is the same as \( \pi \) radians, the sum of opposite angles \( \angle\text{BAD} \) and \( \angle\text{BCD} \) is \( \pi \) radians.

🎯 Exam Tip: Remember the property of cyclic quadrilaterals: opposite angles are supplementary (add up to 180° or \( \pi \) radians).

 

Question 3. 16 right angles in sexagesimal system is equal to
(a) 28° 30' 7"
(b) Option (b)
(c) Option (c)
(d) Option (d)
Answer: (c) Option (c)
In simple words: A right angle is 90 degrees. So, 16 right angles means 16 multiplied by 90 degrees. You then convert this total degree value into degrees, minutes, and seconds, which is the sexagesimal system.

🎯 Exam Tip: Be careful when converting large angles into degrees, minutes, and seconds. Remember that 1 degree has 60 minutes, and 1 minute has 60 seconds.

 

Question 4. How many right angles is equal to 56° 15'?
(a) \( \frac {8}{5} \) right angles
(b) \( \frac {5}{8} \) right angles
(c) \( \frac {3}{5} \) right angles
(d) \( \frac {5}{4} \) right angles
Answer: (b) \( \frac {5}{8} \) right angles
In simple words: First, change 56 degrees and 15 minutes all into degrees by dividing 15 by 60. This gives \( 56.25^\circ \). Then, to find how many right angles this is, divide \( 56.25^\circ \) by \( 90^\circ \). The result is \( \frac{5}{8} \) of a right angle.

🎯 Exam Tip: Convert all parts of an angle (degrees, minutes, seconds) into a single unit (like decimal degrees) before performing calculations. Remember \( 1' = \frac{1}{60}^\circ \).

 

Question 5. In which quadrant will the revolving line lie when it traces an angle \( \frac {1\pi }{ 6 } \)?
(a) 1st quadrant
(b) 2nd quadrant
(c) 3rd quadrant
(d) 4th quadrant
Answer: (a) 1st quadrant
In simple words: An angle of \( \frac{1\pi}{6} \) radians is equal to 30 degrees. This angle is positive, meaning it moves counter-clockwise from the positive x-axis. Since 30 degrees is between 0 and 90 degrees, the revolving line will be in the first quadrant. The first quadrant covers angles from 0 to 90 degrees.

🎯 Exam Tip: It's helpful to remember the degree equivalents for common radian measures (e.g., \( \frac{\pi}{6} = 30^\circ \), \( \frac{\pi}{4} = 45^\circ \), \( \frac{\pi}{3} = 60^\circ \), \( \frac{\pi}{2} = 90^\circ \)).

 

Question 6. 1 radian is equal to:
(a) \( 180^g \)
(b) \( 200^g \)
(c) \( 100^g \)
(d) None of the options
Answer: (b) \( 200^g \)
In simple words: A full circle is 360 degrees, which is also \( 2\pi \) radians or 400 grades. If \( 2\pi \) radians equals 400 grades, then 1 radian is \( \frac{400}{2\pi} \). Since \( \pi \) is approximately 3.14159, 1 radian is about \( 63.66^\circ \), which is 200 grades in the centesimal system. Grades are an alternative unit for angles, where a right angle is 100 grades.

🎯 Exam Tip: Be familiar with different angle measurement systems: degrees (sexagesimal), radians (circular), and grades (centesimal). Know the conversions between them.

 

Question 7. In which quadrant, the angle whose measure is – 400° will lie?
(a) First quadrant
(b) Second quadrant
(c) Third quadrant
(d) Fourth quadrant
Answer: (d) Fourth quadrant
In simple words: A negative angle means we rotate clockwise. An angle of -400 degrees can be found by subtracting full rotations (multiples of 360 degrees). So, -400 degrees is like -360 degrees minus another -40 degrees. This means one full clockwise turn, then an additional 40 degrees clockwise. This takes us into the fourth quadrant.

🎯 Exam Tip: For negative angles, rotate clockwise from the positive x-axis. Subtract or add multiples of 360° to find the coterminal angle within \( 0^\circ \) to \( 360^\circ \).

 

Question 8. The time is taken by the hour hand of a clock in tracing an angle of \( \frac {\pi }{2} \) radian is?
(a) Option (a)
(b) Option (b)
(c) Option (c)
(d) Option (d)
Answer: (b) Option (b)
In simple words: The hour hand moves 360 degrees in 12 hours, which is 30 degrees per hour. An angle of \( \frac{\pi}{2} \) radians is 90 degrees. To find the time taken, divide the total angle by the speed of the hour hand. So, 90 degrees divided by 30 degrees per hour equals 3 hours.

🎯 Exam Tip: Remember that the hour hand moves 360 degrees in 12 hours (or 30 degrees per hour), and the minute hand moves 360 degrees in 60 minutes (or 6 degrees per minute).

 

Question 9. The value of each angle of an equilateral triangle in circular system is:
(a) \( \frac {\pi }{ 2 } \) radian
(b) \( \frac {\pi }{ 3 } \) radian
(c) \( \frac {2\pi }{3} \) radian
(d) \( \pi \) radian
Answer: (b) \( \frac {\pi }{ 3 } \) radian
In simple words: An equilateral triangle has three equal angles, and each angle measures 60 degrees. To convert 60 degrees to radians, multiply it by \( \frac{\pi}{180} \). This calculation gives \( \frac{\pi}{3} \) radians. All angles in an equilateral triangle are always equal.

🎯 Exam Tip: Always remember that the sum of angles in a triangle is 180° or \( \pi \) radians, and for an equilateral triangle, each angle is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

 

Question 10. Each exterior angle of regular octagon is:
(a) \( \frac {\pi }{ 3 } \) radian
(b) \( \frac {\pi }{6} \) radian
(c) \( \frac {\pi }{8} \) radian
(d) \( \frac {\pi }{ 4 } \) radian
Answer: (d) \( \frac {\pi }{ 4 } \) radian
In simple words: A regular octagon has 8 equal sides and 8 equal angles. The sum of all exterior angles of any convex polygon is always 360 degrees. For a regular octagon, divide 360 degrees by 8 to get 45 degrees for each exterior angle. Converting 45 degrees to radians gives \( \frac{\pi}{4} \) radians.

🎯 Exam Tip: For any regular n-sided polygon, each exterior angle is \( \frac{360^\circ}{n} \) or \( \frac{2\pi}{n} \) radians. This is a very useful formula for polygons.

Very Short Answer Type Questions

 

Question 1. Find the magnitude of an interior angle of a regular hexagon
(i) in degree
(ii) in radian
Answer:
(i) To find the interior angle of a regular hexagon in degrees, we use the formula \( \frac{(n-2) \times 180^\circ}{n} \), where n is the number of sides. For a hexagon, \( n = 6 \). So, \( \frac{(6-2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 4 \times 30^\circ = 120^\circ \).
(ii) To find the interior angle in radians, convert 120 degrees using the conversion factor \( \frac{\pi}{180^\circ} \). So, \( 120^\circ = 120 \times \frac{\pi}{180} = \frac{2\pi}{3} \) radians. The radian measure is commonly used in higher mathematics.
In simple words: For a hexagon, each inner angle is 120 degrees. If we change this to radians, it becomes \( \frac{2\pi}{3} \).

🎯 Exam Tip: Remember the formulas for interior and exterior angles of a regular polygon: interior angle \( = \frac{(n-2) \times 180^\circ}{n} \) and exterior angle \( = \frac{360^\circ}{n} \).

 

Question 2. In which quadrant will the revolving ray lie when it makes the following angles.
(i) 280°
(ii) 390°
(iii) – 400°
(iv) – 1120°
Answer:
(i) \( 280^\circ = 3 \times 90^\circ + 10^\circ \). This means it passes through the first three quadrants and enters the fourth quadrant. Hence, the ray will be in the fourth quadrant.
(ii) \( 390^\circ = 4 \times 90^\circ + 30^\circ \). This completes one full rotation (\( 360^\circ \)) and then goes an additional \( 30^\circ \). Therefore, the ray will be in the first quadrant.
(iii) \( -400^\circ = -4 \times 90^\circ - 40^\circ \). This means one full clockwise rotation (\( -360^\circ \)) followed by an additional \( -40^\circ \) clockwise rotation. So, the ray will be in the fourth quadrant.
(iv) \( -1120^\circ = -3 \times 360^\circ - 40^\circ \). This is three full clockwise rotations, followed by an additional \( -40^\circ \) clockwise rotation. The final position will be in the fourth quadrant. Angles are measured from the positive x-axis.
In simple words: (i) 280 degrees is in the fourth part. (ii) 390 degrees (which is 360 + 30) is in the first part. (iii) -400 degrees (which is -360 - 40) is in the fourth part. (iv) -1120 degrees (which is -3 x 360 - 40) is also in the fourth part.

🎯 Exam Tip: To find the quadrant for any angle, add or subtract multiples of \( 360^\circ \) (or \( 2\pi \) radians) until the angle is between \( 0^\circ \) and \( 360^\circ \). For negative angles, rotate clockwise.

 

Question 3. What is the measure of the angle traced out by the hour hand of a clock from 9 a.m. to 3 p.m?
Answer: From 9 a.m. to 3 p.m., there are 6 hours. The hour hand moves \( 30^\circ \) in one hour. So, in 6 hours, it moves \( 6 \times 30^\circ = 180^\circ \). This angle represents exactly two right angles.
In simple words: From 9 a.m. to 3 p.m. is 6 hours. The clock's hour hand moves 30 degrees every hour, so in 6 hours it moves 180 degrees.

🎯 Exam Tip: Remember that the hour hand moves \( 30^\circ \) per hour, and the minute hand moves \( 6^\circ \) per minute. For angles, calculate the total time first.

 

Question 5. Find out whether the angle XOP is positive or negative angle as marked in the following figures.
(i)

(ii)
Answer:
(i) Positive: In this figure, the angle XOP is shown by an arc moving counter-clockwise from the initial ray OX to the terminal ray OP. Counter-clockwise rotation is defined as positive. This is a common way to measure angles in mathematics.
(ii) Negative: In this figure, the angle XOP is shown by an arc moving clockwise from the initial ray OX to the terminal ray OP. Clockwise rotation is defined as negative.
In simple words: (i) The angle XOP is positive because it goes counter-clockwise. (ii) The angle XOP is negative because it goes clockwise.

🎯 Exam Tip: Remember that angles measured counter-clockwise from the positive x-axis are positive, and angles measured clockwise are negative.

 

Question 6. Convert \( \frac {\pi }{72} \) radians into degree (sexagesimal system).
Answer: To convert radians to degrees, we multiply by \( \frac{180^\circ}{\pi} \).
So, \( \frac{\pi}{72} \text{ radians} = \frac{\pi}{72} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{72} = 2.5^\circ \).
Now, we convert the decimal part of the degree to minutes: \( 0.5^\circ = 0.5 \times 60' = 30' \).
Therefore, \( \frac{\pi}{72} \text{ radians} = 2^\circ 30' \). This conversion is important for working with different units of angle measurement.
In simple words: To change \( \frac{\pi}{72} \) radians into degrees, multiply by \( \frac{180}{\pi} \). This gives 2.5 degrees, which is the same as 2 degrees and 30 minutes.

🎯 Exam Tip: Use the conversion factor \( 1 \text{ radian} = \frac{180^\circ}{\pi} \) to convert radians to degrees. Remember that \( 1^\circ = 60' \) (minutes).

Short Answer Type Questions

 

Question 1. Express the following angles in degree, minutes and seconds.
(i) \( \frac {\pi }{8} \) radians
(ii) \( \frac{3\pi}{7} \) radians
Answer:
(i) To convert \( \frac{\pi}{8} \) radians to degrees:
\( \frac{\pi}{8} \text{ radians} = \frac{\pi}{8} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{8} = 22.5^\circ \).
Now, convert the decimal part to minutes: \( 0.5^\circ = 0.5 \times 60' = 30' \).
So, \( \frac{\pi}{8} \text{ radians} = 22^\circ 30' \).
(ii) To convert \( \frac{3\pi}{7} \) radians to degrees:
\( \frac{3\pi}{7} \text{ radians} = \frac{3\pi}{7} \times \frac{180^\circ}{\pi} = \frac{3 \times 180^\circ}{7} = \frac{540^\circ}{7} \approx 77.1428...^\circ \).
This can be written as \( 77^\circ + \frac{1}{7}^\circ \).
Convert \( \frac{1}{7}^\circ \) to minutes: \( \frac{1}{7} \times 60' = \frac{60}{7}' \approx 8.57...'\).
This can be written as \( 8' + \frac{4}{7}' \).
Convert \( \frac{4}{7}' \) to seconds: \( \frac{4}{7} \times 60'' = \frac{240}{7}'' \approx 34.28...'' \).
So, \( \frac{3\pi}{7} \text{ radians} \approx 77^\circ 8' 34'' \). These precise conversions are vital in fields like navigation and astronomy.
In simple words: (i) \( \frac{\pi}{8} \) radians is 22 degrees and 30 minutes. (ii) \( \frac{3\pi}{7} \) radians is about 77 degrees, 8 minutes, and 34 seconds.

🎯 Exam Tip: When converting degrees to minutes and seconds, only convert the decimal part. Multiply the decimal by 60 for minutes, and then the decimal part of the minutes by 60 for seconds.

 

Question 2. Find the length of the arc of a circle of radius 5 cm subtending a central angle measuring 15°.
Answer: First, convert the central angle from degrees to radians:
\( 15^\circ = 15 \times \frac{\pi}{180} \text{ radians} = \frac{\pi}{12} \text{ radians} \).
The radius of the circle is given as \( r = 5 \) cm.
The formula for arc length \( l \) is \( l = r\theta \), where \( \theta \) is the angle in radians.
So, \( l = 5 \times \frac{\pi}{12} = \frac{5\pi}{12} \) cm. This calculation helps in understanding parts of a circle.
In simple words: Change 15 degrees to radians first, which is \( \frac{\pi}{12} \). Then, multiply this by the radius (5 cm) to get the arc length, which is \( \frac{5\pi}{12} \) cm.

🎯 Exam Tip: Always ensure the angle is in radians when using the arc length formula \( l = r\theta \). Convert degrees to radians if necessary before calculation.

 

Question 3. Express the following degrees into grades (i.e., sexagesimal system to centesimal system)
(i) \( 80^\circ \)
(ii) \( 100^\circ \)
Answer: To convert degrees (sexagesimal system) to grades (centesimal system), we use the conversion factor \( 1^\circ = \frac{10}{9} \text{ grades} \). The problem statement shows a worked example for 108 degrees, which we will follow. Let's convert \( 108^\circ \) to grades:
(ii) \( 108^\circ = 108 \times \frac{10}{9} \text{ grades} = 12 \times 10 \text{ grades} = 120^g \). This conversion changes the base for how a right angle is defined.
In simple words: To change degrees to grades, multiply the degrees by \( \frac{10}{9} \). So, 108 degrees becomes 120 grades.

🎯 Exam Tip: Remember that \( 90^\circ = 100^g \) (100 grades), which gives the conversion ratio \( \frac{100}{90} = \frac{10}{9} \) for degrees to grades.

 

Question 4. The vertex angle of an isosceles triangle is 40°. Find the angles of the triangle in radians (i.e. in circular system).
Answer: For an isosceles triangle, the two base angles are equal. Let the vertex angle be \( A = 40^\circ \).
The sum of angles in a triangle is \( 180^\circ \). So, the sum of the two base angles is \( 180^\circ - 40^\circ = 140^\circ \).
Each base angle will be \( \frac{140^\circ}{2} = 70^\circ \).
Now, convert these angles to radians:
Vertex angle \( A = 40^\circ = 40 \times \frac{\pi}{180} = \frac{2\pi}{9} \) radians.
Base angle \( B = 70^\circ = 70 \times \frac{\pi}{180} = \frac{7\pi}{18} \) radians.
Base angle \( C = 70^\circ = 70 \times \frac{\pi}{180} = \frac{7\pi}{18} \) radians. Converting to radians helps in consistent calculations in trigonometry and calculus.
In simple words: If the top angle of an isosceles triangle is 40 degrees, the other two angles at the base are each 70 degrees. In radians, 40 degrees is \( \frac{2\pi}{9} \) and 70 degrees is \( \frac{7\pi}{18} \).

🎯 Exam Tip: For isosceles triangles, the angles opposite the equal sides are equal. Always convert all angles to radians if the question specifies the circular system.

 

Question 5. Find the angles in degrees traced out by the big and small hands of a clock in 10 minutes.
Answer: The small hand (hour hand) of a clock traces 4 right angles in 12 hours.
Angle traced by the hour hand in 12 hours = \( 4 \times 90^\circ = 360^\circ \).
Angle traced by the hour hand in 1 hour = \( \frac{360^\circ}{12} = 30^\circ \).
In 60 minutes, the hour hand traces \( 30^\circ \).
So, in 1 minute, the hour hand traces \( \frac{30^\circ}{60} = 0.5^\circ \).
Therefore, in 10 minutes, the hour hand traces \( 0.5^\circ \times 10 = 5^\circ \).

The big hand (minute hand) of a clock traces \( 360^\circ \) in 60 minutes.
So, in 1 minute, the minute hand traces \( \frac{360^\circ}{60} = 6^\circ \).
Therefore, in 10 minutes, the minute hand traces \( 6^\circ \times 10 = 60^\circ \). These calculations are useful for problems involving relative speeds of clock hands.
In simple words: In 10 minutes, the big hand (minute hand) moves 60 degrees. The small hand (hour hand) moves only 5 degrees in the same time.

🎯 Exam Tip: Remember the speeds: minute hand moves \( 6^\circ \) per minute, hour hand moves \( 0.5^\circ \) per minute (or \( 30^\circ \) per hour). Calculate angles for each hand separately.

 

Question 6. Write the measurement of exterior and interior angles of regular pentagon
(i) in degrees
(ii) in radian
Answer: A regular pentagon has \( n=5 \) sides.
(i) In degrees:
Exterior angle \( = \frac{360^\circ}{n} = \frac{360^\circ}{5} = 72^\circ \).
Interior angle \( = \frac{(n-2) \times 180^\circ}{n} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 3 \times 36^\circ = 108^\circ \).
Alternatively, interior angle \( = 180^\circ - \text{exterior angle} = 180^\circ - 72^\circ = 108^\circ \).
(ii) In radians:
Exterior angle \( = \frac{2\pi}{n} = \frac{2\pi}{5} \) radians.
Interior angle \( = \frac{(n-2)\pi}{n} = \frac{(5-2)\pi}{5} = \frac{3\pi}{5} \) radians.
Alternatively, interior angle \( = \pi - \text{exterior angle} = \pi - \frac{2\pi}{5} = \frac{3\pi}{5} \) radians. These formulas allow quick calculation of angles for any regular polygon.
In simple words: For a regular pentagon, each outside angle is 72 degrees or \( \frac{2\pi}{5} \) radians. Each inside angle is 108 degrees or \( \frac{3\pi}{5} \) radians.

🎯 Exam Tip: Remember the sum of exterior angles of any polygon is \( 360^\circ \) (or \( 2\pi \) radians), and for regular polygons, interior and exterior angles add up to \( 180^\circ \) (or \( \pi \) radians).

 

Question 7. Express 104°30' in radians.
Answer: First, convert 104°30' entirely into degrees:
\( 30' = \frac{30}{60}^\circ = 0.5^\circ \).
So, \( 104^\circ 30' = 104.5^\circ \).
Now, convert degrees to radians using the conversion factor \( \frac{\pi}{180^\circ} \):
\( 104.5^\circ = 104.5 \times \frac{\pi}{180} = \frac{104.5\pi}{180} \).
To simplify the fraction, multiply the numerator and denominator by 10 to remove the decimal:
\( = \frac{1045\pi}{1800} \).
Divide both by 5: \( = \frac{209\pi}{360} \) radians. This method provides an accurate radian measure for the given angle.
In simple words: To change 104 degrees and 30 minutes into radians, first make it 104.5 degrees. Then multiply by \( \frac{\pi}{180} \). This gives \( \frac{209\pi}{360} \) radians.

🎯 Exam Tip: Always convert minutes and seconds to decimal degrees first before converting the entire angle to radians. Use the fraction form to maintain precision.

 

Question 8. Find the angle subtended at the centre by the circumference of the circle.
Answer: Let \( r \) be the radius of the circle.
The circumference of the circle is \( C = 2\pi r \).
We know that the angle \( \theta \) subtended by an arc of length \( l \) at the centre of a circle with radius \( r \) is given by \( \theta = \frac{l}{r} \), where \( \theta \) is in radians.
In this case, the arc length is the entire circumference, so \( l = C = 2\pi r \).
Substitute this into the formula:
\( \theta = \frac{2\pi r}{r} = 2\pi \) radians.
Therefore, the angle subtended at the centre by the entire circumference of the circle is \( 2\pi \) radians, which is equal to \( 360^\circ \). This confirms that a full circle is \( 2\pi \) radians.
In simple words: The entire circle's edge (circumference) makes an angle of \( 2\pi \) radians (or 360 degrees) at the center. This is because \( 2\pi \) radians represents one full turn around a circle.

🎯 Exam Tip: Remember that a full circle is \( 360^\circ \) or \( 2\pi \) radians. The formula \( \theta = \frac{l}{r} \) is only valid when \( \theta \) is expressed in radians.

 

Question 9. The difference between two acute angles of a right-angled triangle is \( \frac {\pi }{9} \). Find the angles in degrees.
Answer: First, convert the given difference from radians to degrees:
\( \frac{\pi}{9} \text{ radians} = \frac{\pi}{9} \times \frac{180^\circ}{\pi} = 20^\circ \).
In a right-angled triangle, one angle is \( 90^\circ \). The sum of the other two acute angles (let's call them \( x \) and \( y \)) must also be \( 90^\circ \).
So, we have two equations:
1. \( x + y = 90^\circ \) (Sum of acute angles in a right triangle)
2. \( x - y = 20^\circ \) (Given difference)
Now, solve these simultaneous equations:
Add (1) and (2):
\( (x + y) + (x - y) = 90^\circ + 20^\circ \)
\( 2x = 110^\circ \)
\( x = \frac{110^\circ}{2} = 55^\circ \).
Substitute \( x = 55^\circ \) into equation (1):
\( 55^\circ + y = 90^\circ \)
\( y = 90^\circ - 55^\circ = 35^\circ \).
Thus, the two acute angles are \( 55^\circ \) and \( 35^\circ \). This problem combines geometry with angle conversions.
In simple words: First, change the given difference of \( \frac{\pi}{9} \) radians to 20 degrees. In a right triangle, the two small angles add up to 90 degrees. If their difference is 20 degrees, then the angles are 55 degrees and 35 degrees.

🎯 Exam Tip: Remember that the sum of angles in a triangle is \( 180^\circ \). For a right-angled triangle, the sum of the two acute angles is \( 90^\circ \). Always convert all values to the required unit (e.g., degrees) before solving.

 

Question 10. Find the angle subtended at the centre of a circle of radius 3 cm by an arc of length 1 cm.
Answer: We are given:
Radius of the circle, \( r = 3 \) cm.
Length of the arc, \( l = 1 \) cm.
The formula for the angle \( \theta \) subtended at the centre by an arc is \( \theta = \frac{l}{r} \), where \( \theta \) is in radians.
Substitute the given values:
\( \theta = \frac{1 \text{ cm}}{3 \text{ cm}} = \frac{1}{3} \) radian.
To express this in degrees, minutes, and seconds, we convert radians to degrees:
\( \frac{1}{3} \text{ radian} = \frac{1}{3} \times \frac{180^\circ}{\pi} = \frac{60^\circ}{\pi} \).
Using \( \pi \approx \frac{22}{7} \):
\( \theta = \frac{60^\circ}{22/7} = \frac{60 \times 7}{22}^\circ = \frac{420}{22}^\circ = \frac{210}{11}^\circ \approx 19.0909...^\circ \).
Now convert to degrees, minutes, and seconds:
\( 19.0909...^\circ = 19^\circ + 0.0909...^\circ \).
\( 0.0909...^\circ \times 60' = \frac{1}{11} \times 60' = \frac{60}{11}' \approx 5.4545...'\).
\( 5.4545...'\) = \( 5' + 0.4545...'\).
\( 0.4545...'\) \( \times 60'' = \frac{5}{11} \times 60'' = \frac{300}{11}'' \approx 27.27...'' \).
So, the angle is approximately \( 19^\circ 5' 27'' \). This illustrates how small arc lengths create small central angles.
In simple words: The angle at the center of the circle is found by dividing the arc length (1 cm) by the radius (3 cm), which gives \( \frac{1}{3} \) radian. This is roughly 19 degrees, 5 minutes, and 27 seconds.

🎯 Exam Tip: Always state the angle in radians unless specifically asked for degrees, minutes, or seconds. Remember that 1 radian is approximately \( 57.3^\circ \).