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Radian and Steradian
A radian is the plane angle at the centre of a circle subtended by an arc whose length is equal to the radius of the circle. A steradian is the solid angle at the centre of a sphere that subtends an area on the spherical surface equal to the square of the sphere’s radius.
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Radian and Steradian
Learning Outcomes
- Define plane angle and solid angle.
- Define radian and steradian.
- Write their correct SI symbols.
- Calculate a plane angle using arc length and radius.
- Calculate a solid angle using spherical surface area and radius.
- Convert radians into degrees and degrees into radians.
- State the angles around a complete circle and sphere.
- Explain why radian and steradian are dimensionless derived units.
- Distinguish clearly between radian and steradian.
Remember: Radian deals with an arc on a circle. Steradian deals with an area on a sphere.
Recommended for Class 11
Radian and Steradian
Chapter 1 — Physical Quantities and Measurements · Topic 6: Radian and Steradian
Learning Outcomes
- Define plane angle and solid angle.
- Define radian and steradian.
- Write their correct SI symbols.
- Calculate a plane angle using arc length and radius.
- Calculate a solid angle using spherical surface area and radius.
- Convert radians into degrees and degrees into radians.
- State the angles around a complete circle and sphere.
- Explain why radian and steradian are dimensionless derived units.
- Distinguish clearly between radian and steradian.
Definition
Radian
Its symbol is: rad
Steradian
Its symbol is: sr The radian and steradian are coherent SI derived units for plane angle and solid angle respectively. NIST SP 330: Plane and Solid Angles
Introduction
An ordinary angle between two lines is called a plane angle. It is measured in radians or degrees. A solid angle describes how large a surface appears from a particular point. It extends in more than one direction and is measured in steradians. For example: The turning of a wheel involves a plane angle. The area covered by the beam of a light source involves a solid angle. The field of view of a camera may be described using angles. Radiation spreading outward from a point source may be described using solid angles. In SI: Plane angle → radian Solid angle → steradian
Key Terms
- Plane angle
- Angle formed between two lines in a plane
- Solid angle
- Angle representing the spread of a surface from a point
- Radian
- SI derived unit of plane angle
- Steradian
- SI derived unit of solid angle
- Arc
- A portion of the circumference of a circle
- Arc length
- Length measured along a circular arc
- Radius
- Distance from the centre to the circular or spherical surface
- Subtend
- To form an angle at a particular point
- Degree
- A non-SI unit commonly used for plane angle
- Dimensionless quantity
- A quantity formed by the ratio of quantities with identical dimensions
Detailed Concept Explanation
Plane angle
Suppose an arc of length s is formed on a circle of radius r. The angle θ at the centre is:
- θ
- Plane angle (rad)
- s
- Arc length (m)
- r
- Radius (m)
When: s = r Then: θ = rr θ = 1 rad Therefore, one radian is formed when the arc length equals the radius.
Complete circle in radians
The circumference of a circle is: C = 2πr The angle around a complete circle is: θ = Cr Substitute the circumference: θ = 2πr / r θ = 2π rad Therefore: 360° = 2π rad Dividing both sides by 2: 180° = π rad
Solid angle
A solid angle is represented by the Greek capital letter omega: Ω If an area A on the surface of a sphere of radius r is subtended at its centre, then:
- Ω
- Solid angle (sr)
- A
- Area on the spherical surface (m²)
- r
- Radius of the sphere (m)
When: A = r² Then: Ω = r²r² Ω = 1 sr Therefore, one steradian is formed when the subtended spherical area equals r².
Complete sphere in steradians
The surface area of a sphere is: A = 4πr² The total solid angle is: Ω = Ar² Substitute the surface area: Ω = 4πr² / r² Ω = 4π sr Therefore: Solid angle around a complete sphere = 4π sr The solid angle of a hemisphere is: Ω = 4π sr / 2 Ω = 2π sr
Dimensionless nature
For a radian: θ = sr Both s and r have the unit metre: [θ] = m / m [θ] = 1 For a steradian: Ω = Ar² Both A and r² have the unit square metre: [Ω] = m² / m² [Ω] = 1 Therefore, radian and steradian are quantities of dimension one. Their symbols are still written when needed to show that the quantities represent plane and solid angles. NIST SP 330: Quantities with Unit One
Concept in Easy Words
A radian tells us how much we have turned around a circle. If the length of a circular arc is exactly equal to the circle’s radius, the angle at the centre is one radian. A steradian tells us how widely an area spreads when viewed from the centre of a sphere. If the area covered on the spherical surface equals the radius squared, the solid angle is one steradian. Remember: Radian deals with an arc on a circle. Steradian deals with an area on a sphere.
Important Facts
- Radian is the coherent SI unit of plane angle.
- Steradian is the coherent SI unit of solid angle.
- The symbol of radian is rad.
- The symbol of steradian is sr.
- Radian and steradian are SI derived units, not SI base units.
- They were once classified as supplementary units, but this separate category was removed in 1995.
- A complete circle contains 2π radians.
- A straight angle contains π radians.
- A right angle contains π/2 radians.
- A complete sphere contains 4π steradians.
- A hemisphere contains 2π steradians.
- Degree is commonly used but is not an SI unit.
- Radians should normally be used in standard angular-motion equations.
- The symbols rad and sr are not pluralized.
Formulas and Mathematical Explanation
Plane angle
- θ
- Plane angle (rad)
- s
- Arc length (m)
- r
- Radius (m)
When to use: Rearranging for arc length
When to use: Rearranging for radius
Solid angle
- Ω
- Solid angle (sr)
- A
- Spherical area (m²)
- r
- Radius (m)
When to use: Rearranging for spherical area
When to use: Rearranging for radius
Radians and degrees
- 2π rad = 360°
- π rad = 180°
- θdegree = θradian × (180° / π rad)
- θradian = θdegree × (π rad / 180°)
Common angular values
| Degrees | Radians |
|---|---|
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Units, Symbols, and Conversions
Quantities and SI units
| Quantity | Quantity symbol | SI unit | Unit symbol |
|---|---|---|---|
| Plane angle | θ | radian | rad |
| Solid angle | Ω | steradian | sr |
| Arc length | s | metre | m |
| Radius | r | metre | m |
| Spherical area | A | square metre | m² |
Useful conversions
- 1 rad = 180° / π
- 1 rad ≈ 57.3°
- 1° = π/180 rad
- 1° ≈ 0.01745 rad
Correct notation: 2 rad 3 sr Incorrect notation: 2 rads 3 srs
Solved Examples
Example 1: Calculating a plane angle
An arc of length 6 m is formed on a circle of radius 3 m. Calculate the angle in radians.
- s = 6 m
- r = 3 m
Formula: θ = sr
θ = 6 m / 3 m
θ = 2 rad
Key step: Divide the arc length by the radius.
Final answer: θ = 2 rad
Example 2: Calculating arc length
A circle has radius 4 m. Find the arc length subtended by an angle of 2.5 rad.
- r = 4 m
- θ = 2.5 rad
Formula: s = rθ
s = (4 m)(2.5)
s = 10 m
Key step: The angle must be expressed in radians when using s = rθ.
Final answer: s = 10 m
Example 3: Degrees to radians
Convert 60° into radians.
- θ = 60°
Formula: θradian = θdegree × (π rad / 180°)
θ = 60° × (π rad / 180°)
θ = 60π / 180 rad
θ = π/3 rad
Key step: Multiply the degree value by π/180.
Final answer: 60° = π/3 rad
Example 4: Radians to degrees
Convert 2.5 rad into degrees.
- θ = 2.5 rad
Formula: θdegree = θradian × (180° / π rad)
θ = 2.5 rad × (180° / π rad)
θ = 450° / π
θ ≈ 143.2°
Key step: Multiply the radian value by 180π.
Final answer: θ ≈ 143.2°
Example 5: Calculating a solid angle
An area of 8 m² on a spherical surface is viewed from the centre of a sphere of radius 2 m. Calculate the solid angle.
- A = 8 m²
- r = 2 m
Formula: Ω = Ar²
Ω = 8 m²2 m²
Ω = 8 m² / 4 m²
Ω = 2 sr
Key step: Square the radius before dividing the area by it.
Final answer: Ω = 2 sr
Example 6: Area subtended by a solid angle
A solid angle of 3 sr is formed at the centre of a sphere of radius 4 m. Find the corresponding spherical area.
- Ω = 3 sr
- r = 4 m
Formula: A = Ω r²
A = (3)(4 m)²
A = (3)(16 m²)
A = 48 m²
Key step: Multiply the solid angle by the square of the radius.
Final answer: A = 48 m²
Real-Life Examples and Applications
Applications of radians
- Angular displacement of rotating wheels
- Circular motion calculations
- Rotational speed and acceleration
- Oscillations and wave equations
- Motion of planets and satellites
- Engineering calculations involving gears
- Trigonometric functions in calculus
Applications of steradians
- Spread of light from a bulb
- Radiation emitted by a star
- Field of view of cameras and telescopes
- Directional sensitivity of microphones
- Radio and satellite antennas
- Luminous intensity and radiant intensity
- Astronomical observations
Comparison Table
| Feature | Radian | Steradian |
|---|---|---|
| Measures | Plane angle | Solid angle |
| Symbol | rad | sr |
| Quantity symbol | θ | Ω |
| Based on | Arc length | Spherical surface area |
| Formula | θ = sr | Ω = Ar² |
| Unit ratio | m/m | m²/m² |
| Complete circle or sphere | 2π rad | 4π sr |
| SI classification | Dimensionless derived unit | Dimensionless derived unit |
| Typical application | Rotation | Spread of light or radiation |
Key Logic
The same central idea is used in both definitions: Angle = quantity subtended / appropriate power of radius For a circle: Plane angle = arc length / radius For a sphere: Solid angle = spherical area / radius² The radius determines the size of the circle or sphere, but the ratio gives an angle independent of that size.
Tips and Tricks
Common Mistakes
- Calling radian an SI base unit → It is an SI derived unit
- Calling steradian a plane angle → Steradian measures solid angle
- Using θ = Ar² → Use Ω = Ar²
- Using Ω = sr → Use θ = sr
- Forgetting to square r for a solid angle → Use r²
- Writing 2 rads → Write 2 rad
- Writing 3 srs → Write 3 sr
- Using degrees directly in s = rθ → Convert degrees into radians first
- Saying a complete sphere contains 2π sr → It contains 4π sr
- Adding arc length and radius without matching units → Convert them to the same unit first
Topic Summary
Radian and steradian are the coherent SI derived units of plane angle and solid angle. A radian is defined using the ratio of arc length to radius, while a steradian is defined using the ratio of spherical area to the square of the radius. One complete circle contains 2π radians, and one complete sphere contains 4π steradians. Although their dimensional ratios reduce to one, the symbols rad and sr are retained to identify the kind of angle being measured.
Radian and Steradian — Exam depth
Exam Facts
- The SI unit of plane angle is radian.
- The SI unit of solid angle is steradian.
- The radian symbol is rad.
- The steradian symbol is sr.
- Radian and steradian are coherent derived units.
- They are quantities with the unit one.
- Degree is not an SI unit, although it is accepted for use.
- One radian is approximately 57.3°.
- A right angle is π/2 rad.
- A straight angle is π rad.
- A complete revolution is 2π rad.
- A complete sphere subtends 4π sr at its centre.
- A hemisphere subtends 2π sr.
Short-Question Preparation
Q. Define radian and steradian.
A. A radian is the plane angle formed at the centre of a circle when the subtended arc length equals the radius. Its symbol is rad. A steradian is the solid angle formed at the centre of a sphere when the subtended spherical area equals the square of the radius. Its symbol is sr. Radian and steradian are dimensionless coherent SI derived units. A complete circle contains 2π rad, while a complete sphere contains 4π sr.
Long-Question Preparation
Q. Explain radian and steradian and derive the total angle around a circle and sphere.
A. A plane angle is the angle formed between two lines in a plane. Its coherent SI unit is the radian. A radian is represented by the symbol rad. Plane angle is defined as the ratio of arc length to radius. Therefore, θ = sr. One radian is formed when the arc length equals the radius. The circumference of a circle is 2πr. Dividing the circumference by the radius gives 2π. Therefore, a complete circle contains 2π rad. A solid angle represents the spread of a surface from a point. Its coherent SI unit is the steradian. A steradian is represented by the symbol sr. Solid angle is defined as spherical area divided by radius squared. Therefore, Ω = Ar². One steradian is formed when the spherical area equals r². The surface area of a sphere is 4πr². Dividing this area by r² gives 4π. Therefore, a complete sphere contains 4π sr. Radian and steradian are dimensionless because their units cancel. Their special symbols are retained to identify plane and solid angles clearly.
Exam Focus
Exam Focus
Important points
- The SI unit of plane angle is radian.
- The SI unit of solid angle is steradian.
- The radian symbol is rad.
- The steradian symbol is sr.
- Radian and steradian are coherent derived units.
- They are quantities with the unit one.
- Degree is not an SI unit, although it is accepted for use.
- One radian is approximately 57.3°.
- A right angle is π/2 rad.
- A straight angle is π rad.
- A complete revolution is 2π rad.
- A complete sphere subtends 4π sr at its centre.
- A hemisphere subtends 2π sr.
Important formulas
- θ = sr
- s = rθ
- Ω = Ar²
- A = Ω r²
- 2π rad = 360°
- π rad = 180°
- Complete sphere = 4π sr
- Hemisphere = 2π sr
Common exam traps
- Calling radian an SI base unit → It is an SI derived unit
- Calling steradian a plane angle → Steradian measures solid angle
- Using θ = Ar² → Use Ω = Ar²
- Using Ω = sr → Use θ = sr
- Forgetting to square r for a solid angle → Use r²
- Writing 2 rads → Write 2 rad
- Writing 3 srs → Write 3 sr
- Using degrees directly in s = rθ → Convert degrees into radians first
- Saying a complete sphere contains 2π sr → It contains 4π sr
- Adding arc length and radius without matching units → Convert them to the same unit first
Long question structure
A plane angle is the angle formed between two lines in a plane. Its coherent SI unit is the radian. A radian is represented by the symbol rad. Plane angle is defined as the ratio of arc length to radius. Therefore, θ = sr. One radian is formed when the arc length equals the radius. The circumference of a circle is 2πr. Dividing the circumference by the radius gives 2π. Therefore, a complete circle contains 2π rad. A solid angle represents the spread of a surface from a point. Its coherent SI unit is the steradian. A steradian is represented by the symbol sr. Solid angle is defined as spherical area divided by radius squared. Therefore, Ω = Ar². One steradian is formed when the spherical area equals r². The surface area of a sphere is 4πr². Dividing this area by r² gives 4π. Therefore, a complete sphere contains 4π sr. Radian and steradian are dimensionless because their units cancel. Their special symbols are retained to identify plane and solid angles clearly.
Common Mistakes
Common Mistakes
- Mistake
Calling radian an SI base unit → It is an SI derived unit
- Mistake
Calling steradian a plane angle → Steradian measures solid angle
- Mistake
Using θ = Ar² → Use Ω = Ar²
- Mistake
Using Ω = sr → Use θ = sr
- Mistake
Forgetting to square r for a solid angle → Use r²
- Mistake
Writing 2 rads → Write 2 rad
- Mistake
Writing 3 srs → Write 3 sr
- Mistake
Using degrees directly in s = rθ → Convert degrees into radians first
- Mistake
Saying a complete sphere contains 2π sr → It contains 4π sr
- Mistake
Adding arc length and radius without matching units → Convert them to the same unit first
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
Plane angle is measured in radians.
Solid angle is measured in steradians.
Radian symbol: rad
Steradian symbol: sr
A radian is the plane angle at the centre of a circle subtended by an arc whose length is equal to the radius of the circle. A steradian is the solid angle at the centre of a sphere that subtends an area on the spherical surface equal to the square of the sphere’s radius.
θ = sr; s = rθ; Ω = Ar²; A = Ω r²; 180° = π rad; 360° = 2π rad; complete sphere = 4π sr
- Definition
- A radian is the plane angle at the centre of a circle subtended by an arc whose length is equal to the radius of the circle. A steradian is the solid angle at the centre of a sphere that subtends an area on the spherical surface equal to the square of the sphere’s radius.
- Formula
- θ = sr; s = rθ; Ω = Ar²; A = Ω r²; 180° = π rad; 360° = 2π rad; complete sphere = 4π sr
- Example
- s = 6 m, r = 3 m → θ = 2 rad; A = 8 m², r = 2 m → Ω = 2 sr
- Critical warning
- Calling radian an SI base unit → It is an SI derived unit
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Practice
Practice
Q1. Define a plane angle.
Practice
Q2. Define one radian.
Practice
Q3. Define one steradian.
Practice
Q4. Why are radian and steradian dimensionless?
Practice
Q5. How many radians are present in a complete circle?
Practice
Q6. How many steradians are present in a complete sphere?
Practice
Q7. Differentiate between radian and steradian.
Practice
Q8. Why must an angle be in radians when using s = rθ?
Practice
Q9. Find the angle when s = 12 m and r = 4 m.
Practice
Q10. Find the arc length when r = 5 m and θ = 3 rad.
Practice
Q11. Convert 45° into radians.
Practice
Q12. Convert π/6 rad into degrees.
Practice
Q13. Find the solid angle when A = 20 m² and r = 2 m.
Practice
Q14. Find the spherical area when Ω = 2 sr and r = 5 m.
Exam Practice
Important Short Question
Q1. Define radian and steradian.
Long Question Preparation
Q2. Explain radian and steradian and derive the total angle around a circle and sphere.