Finding the Difference in Area Between an Inscribed Square and Its Circumscribing Circle

In this article, we will explore the difference in area between a square that is inscribed within a circle and the outer circle itself. The area in question can be determined using known mathematical formulas, making this exploration primarily symbolic reasoning.

Theory Overview

To start, we need to determine the areas of both the circle and the square. Our input will be the diameter of the outer circle. When a square is inscribed in a circle, it touches the circle at each corner. This geometric relationship implies that the diameter of the circle serves as the diagonal of the square.

Recalling the Pythagorean theorem, we know that:

The square of the length of the longest side in a right triangle is equal to the sum of the squares of the other two sides.

Given that our inscribed figure is a square, we can denote the lengths of the two shorter sides as equal. Thus, if d refers to the diameter of the circle, we can express the relationship as follows:

\[ d^2 = 2 \cdot s^2 \]

where s represents the length of one side of the square.

Symbolic Reasoning

To derive the smaller sides, we start from the equation above and manipulate it. We want to eliminate the factor of two on the right-hand side. We can do this by dividing through by 2:

\[ \frac{d^2}{2} = s^2 \]

Consequently, the area of the square can be obtained as:

\[ \text{Area}_{square} = s^2 = \frac{d^2}{2} \]

Calculating the Area of the Circle

Next, we can compute the area of the circle using a formula attributed to Archimedes:

\[ \text{Area}_{circle} = \pi \cdot r^2 \]

Given that the radius r is half of the diameter d, we have:

\[ r = \frac{d}{2} \]

Thus, substituting this into our area formula gives:

\[ \text{Area}_{circle} = \pi \cdot \left(\frac{d}{2}\right)^2 = \frac{\pi \cdot d^2}{4} \]

Combining Areas

Now, we need to combine our findings to determine the difference between the two areas:

\[ \text{Difference} = \text{Area}_{circle} - \text{Area}_{square} \]

Substituting the obtained areas results in:

\[ \text{Difference} = \frac{\pi \cdot d^2}{4} - \frac{d^2}{2} \]

To combine these, we can express 1/2 as 2/4:

\[ \text{Difference} = \frac{\pi \cdot d^2}{4} - \frac{2 \cdot d^2}{4} = \frac{(\pi - 2) \cdot d^2}{4} \]

In APL, this calculation can be represented as follows for an array of diameters:

⍝ Define the function for calculating the area difference
CircleSquareDifference ← {((○ - +⍨) ×⍨ ÷ 4⍨) ⍵}

⍝ Example usage with diameters 2, 4, 6, 8, and 10
CircleSquareDifference 2 4 6 8 10

The output of this APL function will yield the differences for the specified diameters:

1.141592654 4.566370614 10.27433388 18.26548246 28.53981634

Conclusion

This elegant formula reveals the difference in area between an inscribed square and its circumscribing circle. The reasoning process deployed in deducing these results exemplifies the power of symbolic reasoning in mathematics. We can confirm the validity of our conclusions by substituting numeric values for d and observing that the results align with expectations.

Thank you for engaging with this exploration of geometric areas.