The correct answer is Option B: 180/π degree. A radian is a unit of plane angle measurement based on the radius of a circle, where the angle subtended at the center by an arc equal in length to the radius is exactly one radian. Since the circumference of a circle is $2\pi r$, a full rotation of $360^{\circ}$ corresponds to $2\pi$ radians. Step-by-Step Solution & Concepts: A radian is a unit of plane angle measurement based on the radius of a circle, where the angle subtended at the center by an arc equal in length to the radius is exactly one radian. Since the circumference of a circle is $2\pi r$, a full rotation of $360^{\circ}$ corresponds to $2\pi$ radians. Working Principle: The relationship is derived from the definition of a circle's circumference. Since $2\pi \text{ radians} = 360^{\circ}$, dividing both sides by $2\pi$ yields $1 \text{ radian} = \frac{360^{\circ}}{2\pi}$, which simplifies to $\frac{180^{\circ}}{\pi}$. Key Formulas: $1 \text{ rad} = \frac{180^{\circ}}{\pi}$ — Conversion from radians to degrees; $1^{\circ} = \frac{\pi}{180} \text{ rad}$ — Conversion from degrees to radians Key Points: One radian is approximately equal to $57.3^{\circ}$.; Radians are the standard SI unit for plane angle measurement.; The use of radians simplifies calculus operations involving trigonometric functions, such as $\frac{d}{dx}(\sin x) = \cos x$. Advantages: Simplifies expressions in rotation dynamics.; Provides a direct proportional relationship between arc length and radius ($s = r\theta$). Disadvantages: Less intuitive than degrees for architectural and basic geometric construction.; Requires conversion factor $\frac{180}{\pi}$ for practical field measurements. Applications: Angular velocity and acceleration in structural dynamics.; Calculation of strain in structural mechanics.; Signal processing and frequency domain analysis. Additional Info: The value of $\pi$ is approximately 3.14159.. Option B is the standard derivation for converting from circular measure to degree measure. Answer: B is correct — By definition, $2\pi$ radians equal $360^{\circ}$, therefore $1$ radian equals $\frac{360^{\circ}}{2\pi}$, which simplifies to $\frac{180^{\circ}}{\pi}$. Exam Tip: In Structural Mechanics, always verify if your calculator is in 'Degree' or 'Radian' mode before evaluating trigonometric functions, as using the wrong mode is a common source of error in slope and deflection problems.

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CivilStructural Mechanics-II

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1 radian =

180 degree

180/π degree

180/2π degree

360/π degree

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⚙️ TE • Technical Concept & PrincipleCivilStructural Mechanics-II

Option B

180/π degree

Quick Summary:

A radian is a unit of plane angle measurement based on the radius of a circle, where the angle subtended at the center by an arc equal in length to the radius is exactly one radian. Since the circumference of a circle is $2\pi r$ , a full rotation of $360°$ corresponds to $2\pi$ radians.

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⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$1 \text{ rad} = \frac{180°}{\pi}$ — Conversion from radians to degrees

$1° = \frac{\pi}{180} \text{ rad}$ — Conversion from degrees to radians

⚙️ Working Principle

The relationship is derived from the definition of a circle's circumference. Since $2\pi \text{ radians} = 360°$ , dividing both sides by $2\pi$ yields $1 \text{ radian} = \frac{360°}{2\pi}$ , which simplifies to $\frac{180°}{\pi}$ .

📌 Key Points

▸
One radian is approximately equal to $57.3°$ .
▸
Radians are the standard SI unit for plane angle measurement.
▸
The use of radians simplifies calculus operations involving trigonometric functions, such as $\frac{d}{dx}(\sin x) = \cos x$ .

✅ Advantages

▸
Simplifies expressions in rotation dynamics.
▸
Provides a direct proportional relationship between arc length and radius ( $s = r\theta$ ).

❌ Disadvantages / Limitations

▸
Less intuitive than degrees for architectural and basic geometric construction.
▸
Requires conversion factor $\frac{180}{\pi}$ for practical field measurements.

🛠️ Applications / Uses

▸
Angular velocity and acceleration in structural dynamics.
▸
Calculation of strain in structural mechanics.
▸
Signal processing and frequency domain analysis.

📄 Additional Information

▸
The value of $\pi$ is approximately 3.14159.
▸
Option B is the standard derivation for converting from circular measure to degree measure.

📊 Diagram / Illustration

✅

B is correct — By definition, $2\pi$ radians equal $360°$ , therefore $1$ radian equals $\frac{360°}{2\pi}$ , which simplifies to $\frac{180°}{\pi}$ .

Core Concepts Used

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Circular Measure Angular Kinematics Trigonometric Unit Systems

💡 EXAM TIP

In Structural Mechanics, always verify if your calculator is in 'Degree' or 'Radian' mode before evaluating trigonometric functions, as using the wrong mode is a common source of error in slope and deflection problems.

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