A rhombus is inscribed inside a circle, and its perimeter is 24. If the area of the circle is Aπ, find A. 

We are given that a rhombus is inscribed inside a circle, and the perimeter of the rhombus is 24. We need to find the value of \( A \) where the area of the circle is \( A\pi \).

### Step 1: Perimeter of the Rhombus
The perimeter of a rhombus is given by \( 4s \), where \( s \) is the side length of the rhombus. We are told the perimeter is 24, so:
\[
4s = 24 \implies s = 6
\]
Thus, each side of the rhombus is 6 units.

### Step 2: Rhombus Inscribed in a Circle
When a rhombus is inscribed in a circle, the diagonals of the rhombus are diameters of the circle. Also, for a rhombus, the diagonals are perpendicular bisectors of each other. Let the diagonals of the rhombus be \( d_1 \) and \( d_2 \).

The relationship between the diagonals and the side length \( s \) of the rhombus is given by the following formula:
\[
s = \frac{\sqrt{d_1^2 + d_2^2}}{2}
\]
Substituting \( s = 6 \) into the equation:
\[
6 = \frac{\sqrt{d_1^2 + d_2^2}}{2} \implies 12 = \sqrt{d_1^2 + d_2^2} \implies d_1^2 + d_2^2 = 144
\]

### Step 3: Radius of the Circle
The diameter of the circle is the length of one of the diagonals of the rhombus. Since the diagonals are perpendicular bisectors, the circle's radius is half the length of the longer diagonal. However, in this case, we find the radius in terms of \( d_1 \) and \( d_2 \).

The radius \( r \) of the circle is given by half the length of the diagonal:
\[
r = \frac{\sqrt{d_1^2 + d_2^2}}{2} = \frac{12}{2} = 6
\]

### Step 4: Area of the Circle
The area \( A\pi \) of the circle is given by:
\[
\text{Area} = \pi r^2 = \pi \times 6^2 = 36\pi
\]
Thus, \( A = 36 \).

The value of \( A \) is \( \boxed{36} \).

Would you like more details on any step?

Here are 5 related questions for further exploration:
1. How would the problem change if the perimeter of the rhombus was different?
2. Can a square be inscribed in a circle, and what would its area be if its perimeter were known?
3. How does the relationship between the diagonals of a rhombus impact its area?
4. What is the significance of the diagonals of a rhombus in determining the radius of a circumscribed circle?
5. How can you derive the formula for the area of a circle in different contexts?

**Tip:** When a shape is inscribed in a circle, the circle's radius can often be found through symmetry or diagonal relationships within the shape!

A rhombus is inscribed inside a circle, and its perimeter is 24. If the area of the circle is Aπ, find A.

Solve the problem of finding the value of A for a circle where a rhombus with perimeter 24 is inscribed. Learn about rhombus properties, diagonal relationships, and circle area calculations. Suitable for Grades 9-11.

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