Introduction

This article explores how to determine the radius of a circle when its area is equal to that of a square. We will walk you through several examples, explaining the mathematical steps, to help you understand the process and solve similar problems accurately.

The Relationship Between the Area of a Square and a Circle

When the area of a circle is equal to the area of a square, you can derive the radius of the circle using the areas provided. This concept is fundamental in geometry and can be applied in various real-world scenarios.

Example 1: Radius of a Circle with Area Equal to a 7 cm Square

Given the area of a square whose side is 7 cm, we need to find the radius of a circle whose area is equal to that of the square.

Area of the square side2
Area of the square 72 49 cm2

Since the area of the circle is equal to the area of the square, we can use the formula for the area of a circle:

Area of a circle πr2

Setting the areas equal:

πr2 49 cm2

Solving for r:

r2 49/π

Using π ≈ 22/7, we get:

r2 49 * 7/22 343/22 ≈ 15.59

Rounding to two decimal places:

r ≈ 3.95 cm

Example 2: Radius of a Circle with Area Equal to a 4 cm Square

Given a square with a side of 4 cm, we need to find the radius of a circle whose area is equal to that of the square.

Area of the square side2
Area of the square 42 16 cm2

Setting the areas equal:

πr2 16 cm2

Solving for r:

r2 16/π

Using 16 * 7/22, we get:

r2 5.0909/1 π ≈ 5.0909

xtracting the square root:

r ≈ 2.26 cm

Example 3: Radius of a Circle with Area Equal to a 10.5 cm Square

Given a square with a side of 10.5 cm, we need to find the radius of a circle whose area is equal to that of the square.

Area of the square side2
Area of the square 10.52 110.25 cm2

Setting the areas equal:

πr2 110.25 cm2

Solving for r:

r2 110.25/π

Using π ≈ 22/7, we get:

r2 110.25 * 7/22 ≈ 35.079

Rounding to two decimal places:

r ≈ 5.92 cm

Example 4: Radius of a Circle with Area Equal to a 10.5 cm Square (Alternative Method)

Given a square with a side of 10.5 cm, we need to find the radius of a circle whose area is equal to that of the square.

Area of the square side2
Area of the square 10.52 110.25 cm2

Using π ≈ 22/7, we get:

πr2 10.52 110.25 cm2

Solving for r:

r2 110.25/π

Extraction the square root:

r ≈ 5.2 cm

Conclusion

Determining the radius of a circle given that its area is equal to the area of a square requires understanding the relationship between the areas of a circle and a square. By using the formula for the area of a circle and solving for the radius, we can accurately find the radius. Refer to the process outlined in the examples provided to solve similar problems.

For more information on geometry and related concepts, please continue to explore our resources.