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:

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:

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

Solid angle

Radians and degrees

• 2π rad = 360°
• π rad = 180°
• θ_degree = θ_radian × (180° / π rad)
• θ_radian = θ_degree × (π rad / 180°)

Common angular values

Units, Symbols, and Conversions

Quantities and SI units

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

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

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.