Question:medium

All the vertices of a rectangle lie on a circle of radius R. If the perimeter of the rectangle is P, then the area of the rectangle is

Updated On: Jan 15, 2026
  • \(\frac{P^2}{2}−2PR\)
  • \(\frac{P^2}{8}−2R^2\)
  • \(\frac{P^2}{16}−R^2\)
  • \(\frac{P^2}{8}−\frac{R^2}{2}\)
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The Correct Option is B

Solution and Explanation

To determine the area of a rectangle inscribed within a circle, where all vertices lie on the circle, we must consider specific geometric properties. Let the rectangle's side lengths be \(a\) and \(b\). The diagonals of such a rectangle are equivalent to the circle's diameter, which is \(2R\). By applying the Pythagorean theorem, we establish the relationship: \(a^2 + b^2 = (2R)^2 = 4R^2\). The perimeter \(P\) of the rectangle is defined as \(P = 2(a + b)\).
From the perimeter equation, we can derive:
\[a + b = \frac{P}{2}\]
The objective is to find the area \(A\) of the rectangle, represented by the formula:
\[A = a \times b\]
Squaring the sum of the sides expression, \(a + b\), and utilizing the algebraic identity \((a + b)^2 = a^2 + b^2 + 2ab\), we proceed:
\[\left(\frac{P}{2}\right)^2 = 4R^2 + 2ab\]
Rearranging to solve for \(ab\):
\[2ab = \frac{P^2}{4} - 4R^2\]
\[ab = \frac{P^2}{8} - 2R^2\]
Therefore, the area of the rectangle is given by:
\[A = \frac{P^2}{8} - 2R^2\]
This calculation confirms that the correct area is \(\frac{P^2}{8} - 2R^2\).
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In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.
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