The trap: Radian and steradian have named units — so students assume they must have dimensions. They don't. Both are dimensionless ratios. This confusion costs marks on direct recall questions that test whether you understand the difference between "having a unit" and "having dimensions."
What "unit of measurement" means (NCERT Class 11 Physics, Chapter 1, page 1): A unit is a conventionally chosen standard of the same kind as the quantity being measured. Measurement = numerical value × unit. The unit is a reference; the numerical value tells you how many times that reference fits into the measured quantity.
The SI system defines seven base units: metre (length), kilogram (mass), second (time), ampere (electric current), kelvin (temperature), mole (amount of substance), and candela (luminous intensity). Every other unit in physics is derived from combinations of these seven.
Supplementary units — the NEET-relevant subtlety: The SI originally classified radian (plane angle) and steradian (solid angle) as "supplementary units." In 1995, CGPM reclassified them as dimensionless derived units. For NEET purposes, the key fact is: radian = arc length / radius (length / length = dimensionless); steradian = surface area / radius² (length² / length² = dimensionless). They carry unit names for convenience, but their dimensional formula is [M⁰ L⁰ T⁰].
Where NEET tests this: Questions ask whether plane angle and solid angle have dimensions, whether they are the same or different, or whether "having a unit" implies "having dimensions." The correct framing: both are dimensionless; both have units; the unit name does not confer dimensions.
Watch-out: Don't confuse "unitless" with "dimensionless." A quantity can be dimensionless yet still have a named unit (radian, steradian). Conversely, a pure ratio like refractive index is both dimensionless and unitless.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
How many base units are defined in the International System of Units (SI)?
Answer: C. The SI defines exactly seven base units: metre, kilogram, second, ampere, kelvin, mole, and candela (NCERT Class 11 Physics, Chapter 1, page 1).
Why A is wrong: A is wrong because 5 omits two base units (likely mole and candela, which students forget are base rather than derived).
Why B is wrong: B is wrong because 6 omits one base unit — commonly candela, which students mistake for a derived photometric unit.
Why D is wrong: D is wrong because 9 incorrectly counts radian and steradian as base units. They are dimensionless derived units, not base units (trap: unit name ≠ base-unit status).
A light-year is a unit of:
Answer: A. A is correct. A light-year is the distance light travels in vacuum in one year, about 9.46 × 10¹⁵ m. It is used for astronomical distances, where metres give unwieldy numbers.
Why B is wrong: B is wrong because the word 'year' makes it sound like a time interval, but it is a distance: the speed of light multiplied by one year.
Why C is wrong: C is wrong because a speed is a distance per unit time. A light-year is the product of a speed and a time, which leaves a distance.
Why D is wrong: D is wrong because 'light' in the name refers to what travels the distance, not to how bright it is. Luminous intensity is measured in candela.
Which of the following statements about radian and steradian is correct?
Answer: A. Radian (arc/radius) and steradian (area/radius²) are both ratios of like quantities, making them dimensionless [M⁰ L⁰ T⁰]. However, they retain named units (rad, sr) for clarity (NCERT Class 11 Physics, Chapter 1, page 1). This directly tests the unit-vs-dimension distinction.
Why B is wrong: B is wrong because radian and steradian share the same dimensional status: both are dimensionless ratios. There is no asymmetry between them.
Why C is wrong: C is wrong because both are dimensionless — having a named unit does not imply having dimensions (trap: confusing 'has a unit' with 'has dimensions').
Why D is wrong: D is wrong because while both are dimensionless, they do have named units (radian and steradian). 'Unitless' would apply to pure numerical ratios like refractive index.
The dimensional formula of solid angle is:
Answer: B. Solid angle = surface area / radius² = L² / L² = dimensionless = [M⁰ L⁰ T⁰]. The steradian is a named unit for a dimensionless quantity (NCERT Class 11 Physics, Chapter 1, page 1).
Why A is wrong: A is wrong because [M⁰ L¹ T⁰] is the dimensional formula of length. Solid angle is area/area (L²/L²), not a length-like quantity.
Why C is wrong: C is wrong because [M⁰ L² T⁰] represents the dimensions of area. Solid angle is area divided by radius squared — the L² cancels completely.
Why D is wrong: D is wrong because [M⁰ L⁻² T⁰] would represent 1/area. Solid angle is area/radius² where both numerator and denominator have L², so L dimensions cancel to L⁰, not L⁻².
A student claims: "Since radian has a unit, plane angle must have dimensions of length." What is the error in this reasoning?
Answer: D. The student conflates "unit" with "dimension." Radian is defined as arc length / radius = L/L = dimensionless. The named unit exists for convenience, not because the quantity carries dimensions (trap: unit name ≠ dimensions).
Why A is wrong: A is wrong because radian IS a recognised SI unit (formerly supplementary, now a dimensionless derived unit). The issue is not that it lacks a unit, but that having a unit does not confer dimensions.
Why B is wrong: B is wrong because plane angle is dimensionless [M⁰ L⁰ T⁰], not length² or any other dimensional combination. Arc/radius = L/L = 1.
Why C is wrong: C is wrong because dimensional status is system-independent. Plane angle is dimensionless in every unit system (SI, CGS, or any other). Dimensions are properties of the physical quantity, not the unit system.
Which pair consists of one SI base quantity and one dimensionless quantity, the two being linked by a physical constant?
Answer: D. D is correct. Amount of substance is one of the seven SI base quantities and carries the base unit mole. Number of particles is a pure count: dimensionless and unitless. Avogadro's constant N_A is the physical constant linking them, N = n·N_A. No other pair here pairs a base quantity with a dimensionless one.
Why A is wrong: A is wrong because luminous flux is not dimensionless — the lumen is cd·sr, a derived unit with a candela in it. Luminous intensity is indeed a base quantity, but the second half of the pair fails.
Why B is wrong: B is wrong because 'temperature' and 'thermodynamic temperature' name the same base quantity in the same base unit. Neither member is dimensionless, and there is no constant relating them.
Why C is wrong: C is wrong because radian and steradian are both dimensionless derived units. Neither member is a base quantity, so the pair has no base half at all.
Pressure is defined as force per unit area. Express 1 pascal in terms of SI base units.
Answer: B. Pressure = Force / Area = (kg·m·s⁻²) / m² = kg·m⁻¹·s⁻². This is a standard derivation from base units (NCERT Class 11 Physics, Chapter 1).
Why A is wrong: A is wrong because kg·m·s⁻² is the unit of force (newton), not pressure. Pressure requires dividing force by area (m²), which changes the length exponent from +1 to −1.
Why C is wrong: C is wrong because kg·m⁻²·s⁻² has the length exponent as −2. Force is kg·m¹·s⁻², so dividing by m² gives m⁽¹⁻²⁾ = m⁻¹, not m⁻².
Why D is wrong: D is wrong because kg·m²·s⁻² is the unit of energy (joule), not pressure. Pressure = energy/volume, so the exponent is m⁻¹, not m².
The SI unit of electric charge is the coulomb. Express 1 coulomb in terms of SI base units.
Answer: A. Charge = current × time. Ampere (A) is the base unit of current; second (s) is the base unit of time. Therefore 1 C = 1 A·s. This is the simplest derived-unit decomposition using base units (NCERT Class 11 Physics, Chapter 1).
Why B is wrong: B is wrong because A·s⁻¹ would be current/time, which has no standard physical meaning. Charge is current multiplied by time, not divided.
Why C is wrong: C is wrong because A²·s would imply current squared times time. Charge is linearly proportional to current (Q = I·t), not quadratically.
Why D is wrong: D is wrong because kg·A·s⁻² mixes mass into the expression. Charge is purely a product of current and time — no mass dimension is involved.
Get a structured 30-day Mechanics plan and a complete formula booklet — delivered to your inbox instantly.
Given
A question states: "The SI unit of plane angle is radian and the SI unit of solid angle is steradian. What are the dimensions of plane angle and solid angle?"
Required
Determine the dimensional formula for both plane angle and solid angle.
Concept
A unit is a measurement standard; a dimension describes the physical nature of a quantity in terms of base quantities (M, L, T, A, K, mol, cd). A quantity can have a named unit yet be dimensionless if it is a ratio of like quantities.
Formula
• Plane angle θ = arc length / radius = L / L• Solid angle Ω = surface area / radius² = L² / L²
Substitution
• θ = L¹ / L¹ = L⁰ = dimensionless• Ω = L² / L² = L⁰ = dimensionless
Calculation
Both ratios yield L⁰. Since no M, T, or other base dimensions appear in the definitions, the full dimensional formula for both is [M⁰ L⁰ T⁰].
Note: The numerical values of the ratios (e.g. arc/radius for a specific angle) are pure numbers. The "radian" and "steradian" labels are conventional names, not dimensional markers.
Final answer
Both plane angle and solid angle are dimensionless: [M⁰ L⁰ T⁰]. They have named SI units (radian, steradian) but no dimensions.
Common trap
Students see "radian" and "steradian" listed as SI units and conclude the quantities must have dimensions. The trap is confusing "has a unit name" with "has dimensions." A dimensionless quantity CAN have a named unit — the name exists for clarity, not because the quantity carries independent dimensions.
Similar NEET-style question
"Which of the following quantities is dimensionless? (A) Pressure (B) Impulse (C) Solid angle (D) Angular momentum." Answer: (C). Solid angle = area/radius² = L²/L² = dimensionless.
---
Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called a *unit*. The result of a measurement of a physical quantity is expressed by a number (numerical measure) accompanied by a unit.
-- NCERT Class 11 Physics, Ch. 1, p. 1These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Similar Terms
Student treats radian/steradian as having dimensions because they have unit names.
Question asks about dimensions of plane angle, solid angle, or comparison.
Plane angle (radian) and solid angle (steradian) are DIMENSIONLESS — they're ratios (arc/radius for radian; surface-area/r² for steradian). They have unit NAMES for clarity but no dimensions.
More in Units and Measurements: 8 exam traps and mistakes · 3 formulas · 8 question patterns from its other lessons.
Plane angle and solid angle have
treats radian as dimensional
Confuses 'has unit' with 'has dimensions'
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
Test yourself on this topic with real past-paper questions:
Practice this topic →