A

To determine the necessary condition for a block to complete a full circular motion starting from a height \( h \) with a radius of curvature \( r \), we can analyze the forces acting on the block and the energy conservation principle regarding potential and kinetic energy.

1. **Energy Conservation**: At the initial height \( h \), the block possesses potential energy and no kinetic energy. The potential energy can be expressed as:
\[ PE = mgh \]
where \( m \) is the mass of the block, \( g \) is the acceleration due to gravity, and \( h \) is the height.

2. **Kinetic Energy at the Bottom**: When the block reaches the bottom of the circular path, its potential energy is converted into kinetic energy. Hence, the kinetic energy \( KE \) at the bottom will be:
\[ KE = \frac{1}{2} mv^2 \]
where \( v \) is the velocity of the block at the bottom.

3. **Equalizing Energies**: Setting the potential energy at the top equal to the kinetic energy at the bottom gives:
\[ mgh = \frac{1}{2} mv^2 \]
Simplifying this leads to:
\[ v^2 = 2gh \]

4. **Centripetal Force Requirement**: For the block to complete the circular motion, it must maintain a minimum speed at the top of the circle to provide the necessary centripetal force. At the top of the circle, the gravitational force must counteract the centripetal force requirement:
\[ mg = \frac{mv^2}{r} \]

For simplification, we can ignore \( m \) since it cancels out:
\[ g = \frac{v^2}{r} \]
Rearranging gives us:
\[ v^2 = rg \]

5. **Combining Equations**: Now we equate the two expressions for \( v^2 \):
\[ 2gh = rg \]
Simplifying gives:
\[ 2h = r \]
or
\[ h = \frac{r}{2} \]

Therefore, the condition for the block to complete the full circular path is that the height \( h \) must be equal to half the radius of the path, which corresponds to **Option A**.