Plane Angle vs Solid Angle (Definition and Difference)
Physics · Units And Measurements · NEET
A plane angle is the "opening" between two lines in a flat 2D plane, measured as theta = arc / radius (unit: radian). A solid angle is the 3D "cone of view" an area covers as seen from a point, measured as omega = area / radius^2 (unit: steradian). Memory hook: plane = arc over radius (one radius), solid = area over radius squared (two radii) - both are just length ratios, so both are dimensionless but still carry a unit.
Plane angle theta = arc l / radius r (one radius, unit radian). Solid angle omega = surface area A / radius^2 (a cone cut on a sphere, unit steradian). Both are ratios of like lengths, so both are dimensionless.
Your doubts, answered
Are plane angle and solid angle dimensionless or do they have dimensions?
Both are dimensionless. Plane angle = arc / radius = length / length, so the length units cancel. Solid angle = area / radius^2 = (length^2) / (length^2), so those cancel too. Their dimensional formula is [M^0 L^0 T^0]. But being dimensionless does NOT mean having no unit - plane angle still uses the unit radian and solid angle uses steradian. This 'unit but no dimension' point is exactly what NEET 2022 asked.
What is the exact difference between radian and steradian?
Radian is the unit of plane angle (a 2D angle in a flat plane). One radian is the angle where the arc length equals the radius. Steradian is the unit of solid angle (a 3D angle, like a cone spreading from a point). One steradian is the solid angle where the surface area on a sphere equals radius^2. A full circle = 2*pi radians; a full sphere = 4*pi steradians.
How do I find the solid angle formula and its maximum value?
Solid angle omega = surface area (A) on a sphere / radius^2 (r^2). For the whole sphere the total surface area is 4*pi*r^2, so omega = 4*pi*r^2 / r^2 = 4*pi steradians (about 12.57 sr). That is the maximum possible solid angle. Compare with plane angle, whose maximum is 2*pi radians for a full circle.
Are plane angle and solid angle fundamental (base) quantities in SI?
No. In the current SI they are treated as dimensionless derived quantities. Earlier (in your NCERT tables) radian and steradian were listed as the two 'supplementary units', but they are still not base units like metre or kilogram. That is why questions ask which quantities share dimensions with angle - the answer is other dimensionless quantities like strain.
⚠️ The NEET trap ✗ Since plane angle and solid angle are dimensionless, they must have no units either. ✓ They are dimensionless (dimensional formula M^0 L^0 T^0) but they DO have units: radian for plane angle and steradian for solid angle. 'Dimensionless' and 'unitless' are not the same thing. 🧠 Dimensionless does not mean unitless - angle proves it. NEET 2022 tested this exact trap: the answer was 'units but no dimensions'.
Real NEET questions
NEET 2022
Plane angle and solid angle have:
A · Units but no dimensions ✓
B · Dimensions but no units
C · No units and no dimensions
D · Both units and dimensions
Solution: Step 1: Plane angle theta = arc / radius = length / length. The length units cancel, so its dimension is [M^0 L^0 T^0] - dimensionless. Step 2: Solid angle omega = area / radius^2 = length^2 / length^2, also [M^0 L^0 T^0] - dimensionless. Step 3: But each still needs a unit to be reported: plane angle is measured in radian, solid angle in steradian. Step 4: So both have units but no dimensions. Correct option: A.
NEET 2024
The quantities which have the same dimensions as those of solid angle are:
A · Stress and angle
B · Strain and arc
C · Angular speed and stress
D · Strain and angle ✓
Solution: Step 1: Solid angle omega = area / radius^2 is dimensionless: [M^0 L^0 T^0]. Step 2: We need two other dimensionless quantities. Strain = change in length / original length = length / length = dimensionless. Step 3: Plane angle = arc / radius = dimensionless. Step 4: Check the rejects - stress = force/area has dimensions [M L^-1 T^-2] (not dimensionless), arc is a length [L], angular speed is [T^-1]. So only 'strain and angle' are both dimensionless. Correct option: D.
Solved Units And Measurements NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
What is the formula for plane angle and solid angle?
Plane angle theta = arc length / radius = l / r (unit: radian). Solid angle omega = surface area / radius^2 = A / r^2 (unit: steradian).
What is the unit of solid angle?
The steradian (symbol: sr). One steradian is the solid angle that cuts an area equal to r^2 on the surface of a sphere of radius r.
What is the maximum value of a solid angle?
4*pi steradians (about 12.57 sr), because the total surface area of a sphere is 4*pi*r^2, and omega = 4*pi*r^2 / r^2 = 4*pi.
Why are plane angle and solid angle dimensionless?
Because each is a ratio of like quantities. Plane angle is length/length and solid angle is area/area (length^2 / length^2), so all units cancel and the dimensional formula is M^0 L^0 T^0.
Are radian and steradian SI base units?
No. They are dimensionless units. NCERT earlier called them supplementary units, but they are not base units like the metre or second.
