Introduction

This article will guide you through the process of finding the area of a circle that is inscribed in a square with a given area of 2. We will break down the steps involved and provide a detailed explanation to help you understand the concept more clearly.

Understanding the Problem

Given Data

We are given a square with an area of 2 square units. Our task is to find the area of a circle inscribed within this square.

Step-by-Step Solution

Step 1: Find the Side Length of the Square

To find the side length of the square, we use the formula for the area of a square:

Area of a square side2

Given that the area is 2, we can rearrange the formula to find the side length:

side √2

Step 2: Determine the Diameter of the Inscribed Circle

The diameter of the circle inscribed in the square is equal to the side length of the square. Therefore, the diameter (d) is equal to √2.

Step 3: Calculate the Radius of the Circle

The radius (r) of the circle is half of the diameter. Hence:

r d/2 √2/2

Step 4: Find the Area of the Circle

To find the area (A) of the circle, we use the area formula:

A πr2

Substituting the value of the radius:

A π(√2/2)2 π(2/4) π(1/2) π/2

Thus, the area of the inscribed circle is π/2 square units.

Verification and Alternative Methods

Let's verify the solution using an alternate method:

Alternative Method 1: Direct Calculation

A square of area 2 has a side length of √2. The radius of the inscribed circle is half of the side length, which is √2/2. Using the area formula for a circle:

A π(√2/2)2 π(2/4) π/2

Alternative Method 2: Using Diagonal Properties

If the length of the side of a square is x, then the area is x2 2. Solving for x, we get:

x √2

The diagonal of the square is √(x2 x2) √(2x2) √(4) 2 units.

The diameter of the inscribed circle is equal to the diagonal of the square, which is 2 units. Therefore, the radius is 1 unit. Using the area formula:

A π(1)2 π

Note that this method doesn't give the correct area for the circle inscribed in the square since the radius is different from the direct calculation method.

Conclusion

By following the steps outlined above, we can confidently state that the area of the inscribed circle in a square with an area of 2 is π/2 square units. This exercise highlights the importance of understanding geometric relationships and applying them correctly.