"

What is the Area of the Largest Circle That Can Be Drawn Inside a Square of Side 14 cm?

" "

To determine the area of the largest circle that can fit inside a square with a side length of 14 cm, we need to understand the relationship between the dimensions of the square and the circle.

" "

Understanding the Largest Inscribed Circle

" "

The largest circle that can fit inside a square is known as the inscribed circle, or incircle. The diameter of this circle is equal to the side length of the square. In this case, the side length of the square is 14 cm, making the diameter of the circle also 14 cm.

" "

Calculating the Radius of the Circle

" "

The radius of the circle is half the diameter. Therefore, the radius ( r ) can be calculated as:

" "

r frac{text{diameter}}{2} frac{14 text{ cm}}{2} 7 text{ cm}

" "

Calculating the Area of the Circle

" "

The area ( A ) of a circle is given by the formula:

" "

A pi r^2

" "

Substituting the radius, we get:

" "

A pi (7 text{ cm})^2 pi times 49 text{ cm}^2 approx 153.94 text{ cm}^2

" "

Therefore, the area of the largest circle that can be drawn inside a square with a side length of 14 cm is approximately 153.94 cm2.

" "

Generalization for Any Square Side Length

" "

For a given square with a side length ( s ), the largest circle inscribed within that square will have a diameter equal to ( s ). The radius of this circle will be ( frac{s}{2} ). The area of this circle can be calculated using the formula:

" "

A pi left(frac{s}{2}right)^2 pi frac{s^2}{4}

" "

Ratio of Areas Between Circle and Square

" "

The area of the square is ( s^2 ). The area of the inscribed circle is ( pi frac{s^2}{4} ). The ratio of the areas of the circle to the square is:

" "

frac{pi frac{s^2}{4}}{s^2} frac{pi}{4}

" "

Expressing this ratio in decimal form:

" "

frac{pi}{4} approx 0.7857

" "

This means that the area of the circle is approximately 78.57% of the area of the square.

" "

Real-world Application

" "

Larger squares or circles can be applied in various real-world scenarios. For example, in architecture, these concepts can be used to design spaces or structures that incorporate circles inscribed within squares, optimizing space usage and aesthetic appeal.