The least perimeter of a cyclic quadrilateral of given area A square units, is

Options

A. `sqrt(A)`
B. `2sqrt(A)`
C. `3sqrt(A)`
D. `4sqrt(A)`

Answer

To find the least perimeter of a cyclic quadrilateral with a given area \( A \) square units, we can follow these steps:

### Step 1: Define the sides and perimeter
Let the sides of the cyclic quadrilateral be \( a, b, c, d \). The perimeter \( P \) of the quadrilateral can be expressed as:
\[ P = a + b + c + d \]

### Step 2: Define the semi-perimeter
The semi-perimeter \( s \) is given by:
\[ s = \frac{P}{2} = \frac{a + b + c + d}{2} \]

### Step 3: Use the formula for the area of a cyclic quadrilateral
The area \( A \) of a cyclic quadrilateral can be calculated using Brahmagupta's formula:
\[ A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \]

### Step 4: Apply the AM-GM inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we can state:
\[ \frac{(s-a) + (s-b) + (s-c) + (s-d)}{4} \geq \sqrt[4]{(s-a)(s-b)(s-c)(s-d)} \]
This implies:
\[ \frac{2s}{4} \geq \sqrt[4]{(s-a)(s-b)(s-c)(s-d)} \]
Thus:
\[ \frac{s}{2} \geq \sqrt[4]{(s-a)(s-b)(s-c)(s-d)} \]

### Step 5: Relate area and perimeter
Substituting the area \( A \) into the inequality:
\[ \frac{s}{2} \geq \sqrt[4]{A} \]
This can be rearranged to:
\[ s \geq 2\sqrt{A} \]

### Step 6: Express perimeter in terms of area
Since \( P = 2s \), we can write:
\[ P \geq 4\sqrt{A} \]

### Conclusion
Thus, the least perimeter of a cyclic quadrilateral with a given area \( A \) is:
\[ \boxed{4\sqrt{A}} \]