RD Sharma Class 11 Solutions for Chapter 4: Measurement of Angles
RD Sharma Class 11 Solutions for Chapter 4: Measurement of Angles are an essential resource for students aiming to master the key concepts of angle measurement in the Class 11 Maths syllabus. This chapter, “Measurement of Angles,” introduces students to the foundational ideas of how angles are measured, the different systems used, and the importance of these concepts in trigonometry and geometry-making it a vital chapter for success in mathematics and competitive exams.
In RD Sharma Solutions for Class 11 Maths Chapter 4, you’ll find step-by-step solutions to all exercises, including detailed explanations for converting between degrees, radians, and grades. These solutions help clarify the difference between the sexagesimal (degree), centesimal (grade), and circular (radian) systems, and guide you through the logic behind each type of angle measurement. Each solution is designed to make even the most challenging problems approachable and easy to understand.
Student’s searching for a downloadable PDF of RD Sharma Chapter 4 Measurement of Angles, you’ll find comprehensive answers, solved examples, and extra questions to reinforce your understanding. The solutions cover all the important topics in the chapter, including the definition of an angle, systems of measurement, conversion formulas, and real-life applications such as finding the angle between clock hands or calculating arc length.
How to solve angle conversion problems in RD Sharma Class 11 Measurement of Angles? Our solutions walk you through the process, ensuring you can confidently convert between degrees, radians, and grades, and apply these concepts in practical situations. You’ll also find guidance on the relationship between angles and real numbers, and see how to solve application-based questions step by step.
Download RD Sharma Solutions for Class 11 Maths Chapter 3 Functions PDF
For those needing help with standard problems and exam-focused questions, RD Sharma Class 11 Measurement of Angles solved examples and exercises providing visual explanations and practical tips. The content is fully aligned with the latest CBSE Class 11 Maths Chapter 4 syllabus, so you can be sure you’re preparing with the most relevant material.
RD Sharma Solutions Class 11 Chapter 4: Measurement of Angles - Step by Step Solutions
Q1.Convert 60° into radians.
Step 1: Recall the conversion formula from degrees to radians.
Step 2: Substitute 60° into the formula.
Radians = 60 × (π/180)
Step 3: Simplify the fraction.
60/180 = 1/3
Step 4: Complete the multiplication.
Radians = 1/3 × π = π/3
Answer: π/3 radians
Q2.Convert π/6 radians to degrees.
Step 1: Recall the conversion formula from radians to degrees.
Step 2: Substitute π/6 into the formula.
Degrees = (π/6) × (180/π)
Step 3: Cancel out π in numerator and denominator.
Degrees = 180/6
Step 4: Simplify the fraction.
180/6 = 30
Answer: 30 degrees
Q3.Convert 1 radian into degrees, minutes, and seconds.
Step 1: Convert radians to degrees using the formula.
Degrees = 1 × (180/π)
Step 2: Calculate the decimal value.
180/π ≈ 57.2958 degrees
Step 3: Separate the whole degrees.
Whole degrees = 57°
Step 4: Calculate the minutes from the decimal part.
Decimal part = 0.2958 × 60 = 17.748 minutes
Step 5: Separate whole minutes and calculate seconds.
Whole minutes = 17'
Decimal part = 0.748 × 60 = 44.88 seconds ≈ 45"
Answer: 57° 17′ 45″
Q4.Convert 150° into radians.
Step 1: Use the conversion formula from degrees to radians.
Step 2: Substitute 150° into the formula.
Radians = 150 × (π/180)
Step 3: Simplify the fraction by dividing by common factor 30.
150/180 = 5/6
Step 4: Complete the calculation.
Radians = 5/6 × π = 5π/6
Answer: 5π/6 radians
Q5.Convert 3π/4 radians to degrees.
Step 1: Use the conversion formula from radians to degrees.
Step 2: Substitute 3π/4 into the formula.
Degrees = (3π/4) × (180/π)
Step 3: Cancel out π in numerator and denominator.
Degrees = (3/4) × 180
Step 4: Calculate the result.
3 × 180 ÷ 4 = 540 ÷ 4 = 135
Answer: 135 degrees
Q6.What is the degree measure of a full circle?
Step 1: Recall the definition of a full circle.
A full circle is one complete rotation around a center point.
Step 2: Recall the standard measure for a full circle.
By definition, a full circle measures 360°.
Answer: 360 degrees
Q7.What is the radian measure of a full circle?
Step 1: Use the conversion formula to convert 360° to radians.
Step 2: Substitute 360° into the formula.
Radians = 360 × (π/180)
Step 3: Simplify the fraction.
360/180 = 2
Step 4: Complete the calculation.
Radians = 2 × π = 2π
Answer: 2π radians
Q8.Find the angle in radians subtended by an arc of length 11 cm at the center of a circle with radius 7 cm.
Step 1: Recall the formula relating arc length, radius, and angle.
where θ is the angle in radians, s is arc length, and r is radius
Step 2: Identify the given values.
Arc length (s) = 11 cm
Radius (r) = 7 cm
Step 3: Substitute values into the formula.
θ = 11/7
Answer: 11/7 radians
Q9.Find the arc length of a sector with radius 10 cm and angle π/3 radians.
Step 1: Recall the formula for arc length.
where s is arc length, r is radius, and θ is angle in radians
Step 2: Identify the given values.
Radius (r) = 10 cm
Angle (θ) = π/3 radians
Step 3: Substitute values into the formula.
s = 10 × (π/3)
Step 4: Calculate the result.
s = 10π/3 cm
Answer: 10π/3 cm
Q10.Find the area of a sector of a circle with radius 6 cm and angle 60°.
Step 1: Convert the angle from degrees to radians.
60° × (π/180) = 60π/180 = π/3 radians
Step 2: Recall the formula for sector area.
where A is area, r is radius, and θ is angle in radians
Step 3: Identify the values.
Radius (r) = 6 cm
Angle (θ) = π/3 radians
Step 4: Substitute values into the formula.
A = ½ × 6² × (π/3)
Step 5: Calculate step by step.
A = ½ × 36 × (π/3)
A = 18 × (π/3)
A = 18π/3 = 6π
Answer: 6π cm²
Q11.Convert 270° into radians.
Step 1: Use the conversion formula from degrees to radians.
Step 2: Substitute 270° into the formula.
Radians = 270 × (π/180)
Step 3: Simplify the fraction by dividing by common factor 90.
270/180 = 3/2
Step 4: Complete the calculation.
Radians = 3/2 × π = 3π/2
Answer: 3π/2 radians
Q12.Convert 7π/6 radians into degrees.
Step 1: Use the conversion formula from radians to degrees.
Step 2: Substitute 7π/6 into the formula.
Degrees = (7π/6) × (180/π)
Step 3: Cancel out π in numerator and denominator.
Degrees = (7/6) × 180
Step 4: Calculate the result.
7 × 180 ÷ 6 = 1260 ÷ 6 = 210
Answer: 210 degrees
Q13.Find the length of an arc that subtends a central angle of 2 radians in a circle of radius 4 cm.
Step 1: Recall the formula for arc length.
where s is arc length, r is radius, and θ is angle in radians
Step 2: Identify the given values.
Radius (r) = 4 cm
Angle (θ) = 2 radians
Step 3: Substitute values into the formula.
s = 4 × 2
Step 4: Calculate the result.
s = 8 cm
Answer: 8 cm
Q14.Convert 1° into radians.
Step 1: Use the conversion formula from degrees to radians.
Step 2: Substitute 1° into the formula.
Radians = 1 × (π/180)
Step 3: Simplify the fraction.
Radians = π/180
Answer: π/180 radians
Q15.Convert 1 radian to degrees (approximate).
Step 1: Use the conversion formula from radians to degrees.
Step 2: Substitute 1 radian into the formula.
Degrees = 1 × (180/π)
Step 3: Calculate using π ≈ 3.14159.
180/3.14159 ≈ 57.2958
Answer: 57.3° (approx)
Q16.Find the angle in radians for arc length 33 cm in a circle of radius 21 cm.
Step 1: Recall the formula relating arc length, radius, and angle.
where θ is the angle in radians, s is arc length, and r is radius
Step 2: Identify the given values.
Arc length (s) = 33 cm
Radius (r) = 21 cm
Step 3: Substitute values into the formula.
θ = 33/21
Step 4: Simplify the fraction by dividing by common factor 3.
33/21 = 11/7
Answer: 11/7 radians
Q17.Find the area of a sector of radius 7 cm and angle 2π/3.
Step 1: Recall the formula for sector area.
where A is area, r is radius, and θ is angle in radians
Step 2: Identify the given values.
Radius (r) = 7 cm
Angle (θ) = 2π/3 radians
Step 3: Substitute values into the formula.
A = ½ × 7² × (2π/3)
Step 4: Calculate step by step.
A = ½ × 49 × (2π/3)
A = 49/2 × (2π/3)
A = 49 × (π/3)
A = 49π/3
Answer: 49π/3 cm²
Q18.A sector has an arc length 15 cm and radius 5 cm. Find the angle in degrees.
Step 1: First find the angle in radians using the formula.
Step 2: Substitute values.
θ = 15/5 = 3 radians
Step 3: Convert radians to degrees.
Step 4: Substitute the value.
Degrees = 3 × (180/π)
Step 5: Calculate using π ≈ 3.14159.
540/3.14159 ≈ 171.89°
Answer: 171.89°
Q19.Convert 2.5 radians into degrees.
Step 1: Use the conversion formula from radians to degrees.
Step 2: Substitute 2.5 radians into the formula.
Degrees = 2.5 × (180/π)
Step 3: Calculate the result.
2.5 × 180 = 450
Step 4: Divide by π.
450/π ≈ 450/3.14159 ≈ 143.24°
Answer: 143.24°
Q20.Convert 1.25 radians into degrees.
Step 1: Use the conversion formula from radians to degrees.
Step 2: Substitute 1.25 radians into the formula.
Degrees = 1.25 × (180/π)
Step 3: Calculate the result.
1.25 × 180 = 225
Step 4: Divide by π.
225/π ≈ 225/3.14159 ≈ 71.62°
Answer: 71.62°
Q21.Convert 330° into radians.
Step 1: Use the conversion formula from degrees to radians.
Step 2: Substitute 330° into the formula.
Radians = 330 × (π/180)
Step 3: Simplify the fraction by dividing by common factor 30.
330/180 = 11/6
Step 4: Complete the calculation.
Radians = 11/6 × π = 11π/6
Answer: 11π/6 radians
Q22.Find the arc length for angle π/2 in a circle of radius 14 cm.
Step 1: Recall the formula for arc length.
where s is arc length, r is radius, and θ is angle in radians
Step 2: Identify the given values.
Radius (r) = 14 cm
Angle (θ) = π/2 radians
Step 3: Substitute values into the formula.
s = 14 × (π/2)
Step 4: Calculate the result.
s = 14π/2 = 7π cm
Answer: 7π cm
Q23.Find the area of sector for a 45° angle in a circle of radius 10 cm.
Step 1: Convert the angle from degrees to radians.
45° × (π/180) = 45π/180 = π/4 radians
Step 2: Recall the formula for sector area.
where A is area, r is radius, and θ is angle in radians
Step 3: Identify the values.
Radius (r) = 10 cm
Angle (θ) = π/4 radians
Step 4: Substitute values into the formula.
A = ½ × 10² × (π/4)
Step 5: Calculate step by step.
A = ½ × 100 × (π/4)
A = 50 × (π/4)
A = 50π/4 = 25π
Answer: 25π cm²
Q24.Convert 90° into radians.
Step 1: Use the conversion formula from degrees to radians.
Step 2: Substitute 90° into the formula.
Radians = 90 × (π/180)
Step 3: Simplify the fraction by dividing by common factor 90.
90/180 = 1/2
Step 4: Complete the calculation.
Radians = 1/2 × π = π/2
Answer: π/2 radians
Q25.Find the angle (in radians) subtended by an arc of length 5 cm in a circle of radius 2 cm.
Step 1: Recall the formula relating arc length, radius, and angle.
where θ is the angle in radians, s is arc length, and r is radius
Step 2: Identify the given values.
Arc length (s) = 5 cm
Radius (r) = 2 cm
Step 3: Substitute values into the formula.
θ = 5/2
Step 4: Calculate the result.
θ = 2.5 radians
Answer: 2.5 radians
FAQs: RD Sharma Class 11 Solutions for Measurement of Angles
1. What are the three main systems of measuring angles in Class 11 Maths?
The three primary systems of measuring angles introduced in Class 11 Maths are:
Sexagesimal System (Degree Measure) – Angles are measured in degrees (°), minutes (′), and seconds (″).
Centesimal System (Grade Measure) – Angles are measured in grades (g), where a right angle equals 100 grades.
Circular System (Radian Measure) – Angles are measured in radians, where the angle subtended at the center of a circle by an arc equal in length to the radius is 1 radian.
2. How do I convert between degrees and radians in RD Sharma?
You can use the following standard formulas for conversions:
Degrees to Radians:
Radians = Degrees × (π/180)
Radians to Degrees:
Degrees = Radians × (180/π)
These formulas are extensively used in RD Sharma's Class 11 Chapter 4 problems and examples.
3. What is the significance of the radian in higher mathematics?
The radian is a natural unit of angular measure in mathematics because it relates arc length directly to radius without additional scaling. It simplifies calculus operations, especially derivatives and integrals involving trigonometric functions, and is used universally in advanced math, physics, and engineering due to its unit-free property.
4. Where can I get step-by-step solutions for RD Sharma Chapter 4 Measurement of Angles?
You can find detailed step-by-step solutions to RD Sharma Class 11 Chapter 4 from:
Educational platforms like Infinity Learn and You can also refer to personalized walkthroughs and solved examples provided by your teacher or coaching notes.
5. Are RD Sharma Chapter 4 solutions as per the latest CBSE syllabus?
Yes, most popular platforms and RD Sharma's latest editions are regularly updated to match the NCERT and CBSE Class 11 syllabus for 2024–25. The chapter "Measurement of Angles" aligns with the curriculum prescribed for Mathematics and is essential for foundational understanding in trigonometry.