Steradian is the solid angle subtended at the centre of the sphere by its surface whose area is equal to ____ of the sphere.

Options

A. the radius
B. square of the radius
C. cube of the radius
D. reciprocal of the radius

Answer

To solve the question regarding the steradian and its relation to the surface area of a sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Steradian**:
- A steradian is a unit of solid angle. It is defined as the angle subtended at the center of a sphere by a surface area on the sphere that is equal to the square of the radius of the sphere.

2. **Surface Area of a Sphere**:
- The formula for the surface area \( A \) of a sphere with radius \( r \) is given by:
\[ A = 4\pi r^2 \]

3. **Definition of Steradian**:
- By definition, if a surface area \( A \) on the sphere is equal to \( r^2 \) (the square of the radius), then the solid angle \( \Omega \) in steradians can be expressed as:
\[ \Omega = \frac{A}{r^2} \]
- Substituting \( A = r^2 \):
\[ \Omega = \frac{r^2}{r^2} = 1 \text{ steradian} \]

4. **Total Solid Angle in a Sphere**:
- The total solid angle around a point in three-dimensional space is \( 4\pi \) steradians. This corresponds to the entire surface area of the sphere.

5. **Conclusion**:
- Therefore, the steradian is the solid angle subtended at the center of the sphere by a surface whose area is equal to the square of the radius of the sphere. Thus, the answer to the question is:
\[ \text{The area is equal to the square of the radius of the sphere.} \]

### Final Answer:
The area is equal to the **square of the radius** of the sphere.

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