The angles of a quadrilateral are in AP, and the greatest angle is double the least. Express the least angle in radians.

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the least angle of the quadrilateral be \( x \).
- Since the angles are in arithmetic progression (AP), we can express the four angles as:
- Least angle: \( x \)
- Second angle: \( x + d \)
- Third angle: \( x + 2d \)
- Greatest angle: \( x + 3d \)
- Here, \( d \) is the common difference.

2. **Use the Given Condition:**
- According to the problem, the greatest angle is double the least angle:
\[ x + 3d = 2x \]
- Rearranging this gives:
\[ 3d = 2x - x \implies 3d = x \implies d = \frac{x}{3} \]

3. **Sum of Angles in a Quadrilateral:**
- The sum of the angles in a quadrilateral is \( 360^\circ \):
\[ x + (x + d) + (x + 2d) + (x + 3d) = 360^\circ \]
- Simplifying this, we get:
\[ 4x + (0 + d + 2d + 3d) = 360^\circ \implies 4x + 6d = 360^\circ \]

4. **Substitute \( d \) in the Sum Equation:**
- Substitute \( d = \frac{x}{3} \) into the equation:
\[ 4x + 6\left(\frac{x}{3}\right) = 360^\circ \]
- This simplifies to:
\[ 4x + 2x = 360^\circ \implies 6x = 360^\circ \implies x = 60^\circ \]

5. **Convert to Radians:**
- To express the least angle in radians, we convert \( 60^\circ \) to radians:
\[ x = 60^\circ = \frac{\pi}{3} \text{ radians} \]

### Final Answer:
The least angle in radians is \( \frac{\pi}{3} \).