In measurement science, the radian is the SI unit used to measure which of the following physical quantities?

Introduction / Context:
Angles are fundamental in geometry, trigonometry, and physics, and there are several ways to measure them. Degrees are common in everyday usage, but in mathematics and physics the radian is the preferred unit because it simplifies many formulas. The question checks whether you know which physical quantity is measured in radians according to the SI system. Recognising this helps you interpret trigonometric functions, wave equations, and rotational motion correctly.

Given Data / Assumptions:

• The unit in question is the radian.
• Options for the measured quantity include temperature, intensity of flame, angle, and solid angle.
• We assume the standard SI usage where radians appear in formulas involving circular motion and periodic functions.
• Solid angle has its own unit, which is related but different.

Concept / Approach:
A radian is defined based on the geometry of a circle. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. Mathematically, a full circle corresponds to 2 * pi radians, which is equal to 360 degrees. Therefore, radian is a unit of plane angle, not of temperature or intensity. Solid angle, which measures the size of an object as seen from a point in three dimensional space, is measured in steradians, not radians. Thus, among the given options, angle is the correct physical quantity measured in radians.

Step-by-Step Solution:
Step 1: Recall that radian is defined using arc length and radius in a circle.Step 2: Recognise that this definition clearly refers to a plane angle at the centre of the circle.Step 3: Note that temperature is measured in kelvin or degrees Celsius, not in radians.Step 4: Understand that intensity of flame could be measured in power units or brightness units, not radians.Step 5: Identify that solid angle uses steradian as its SI unit, leaving angle as the only correct choice for radian.

Verification / Alternative check:
Consider standard mathematical formulas such as the trigonometric expansions sin x and cos x, where x is in radians for the series to hold directly. In physics, formulas for simple harmonic motion, wave propagation, and rotational kinematics also use angular quantities in radians. For example, in circular motion, the relation between arc length s, radius r, and angle theta is s = r * theta, where theta must be in radians. No similar equations use radians for temperature or intensity. Solid angle equations, such as those involving flux passing through an area, specifically use steradians. These examples confirm that radians are units of plane angle.

Why Other Options Are Wrong:
Temperature is a measure of hotness or coldness and has SI unit kelvin, so option A is incorrect. Intensity of flame is related to energy output or brightness and does not use radians as a unit, making option B wrong. Solid angle has its own SI unit called steradian, which is different from radian, so option D is incorrect. Only option C, angle, correctly matches the radian as a unit of measurement.

Common Pitfalls:
Students sometimes confuse radians and steradians because they sound similar. Remember that a radian describes a plane angle in two dimensional geometry, while a steradian describes a solid angle in three dimensional space. Another mistake is to rely only on degrees and think of radians as an abstract concept. Practising conversions between degrees and radians and using formulas where radians appear naturally will solidify your understanding that radians measure ordinary plane angles.

Final Answer:
Angle