Diameter Of Circumcircle Of Rectangle Given Area And Obtuse Angle Between Diagonals Calculator

Formula Used:

\[ D_c = \frac{\sqrt{A \cdot \cot\left(\frac{\pi - \angle_{d(Obtuse)}}{2}\right)}}{\cos\left(\frac{\pi - \angle_{d(Obtuse)}}{2}\right)} \]

Diameter of Circumcircle:

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1. What is the Diameter of Circumcircle of Rectangle?

The diameter of the circumcircle of a rectangle is the diameter of the circle that passes through all four vertices of the rectangle. For any rectangle, the circumcircle's center is at the intersection point of the diagonals, and its diameter equals the length of the diagonal of the rectangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ D_c = \frac{\sqrt{A \cdot \cot\left(\frac{\pi - \angle_{d(Obtuse)}}{2}\right)}}{\cos\left(\frac{\pi - \angle_{d(Obtuse)}}{2}\right)} \]

Where:

\( D_c \) — Diameter of circumcircle (m)
\( A \) — Area of rectangle (m²)
\( \angle_{d(Obtuse)} \) — Obtuse angle between diagonals (radians)
\( \pi \) — Mathematical constant pi (approximately 3.14159)

Explanation: This formula derives from the geometric relationships between the rectangle's area, the angle between its diagonals, and the properties of the circumscribed circle.

3. Importance of Circumcircle Diameter Calculation

Details: Calculating the circumcircle diameter is important in various geometric applications, construction projects, and engineering designs where circular elements need to encompass rectangular components or where the relationship between rectangular and circular geometries needs to be determined.

4. Using the Calculator

Tips: Enter the area of the rectangle in square meters and the obtuse angle between diagonals in degrees (must be between 90° and 180°). All values must be positive and valid.

5. Frequently Asked Questions (FAQ)

Q1: Why is the angle between diagonals important?
A: The angle between diagonals determines the shape of the rectangle. Different angles correspond to rectangles with different aspect ratios but the same area.

Q2: What is the range of valid angles?
A: The obtuse angle between diagonals must be between 90° and 180°. At 90°, the rectangle becomes a square, and at 180°, it degenerates into a line.

Q3: How is this related to the diagonal length?
A: The diameter of the circumcircle is exactly equal to the length of the diagonal of the rectangle.

Q4: Can this formula be used for squares?
A: Yes, for a square, the angle between diagonals is 90°, and the formula simplifies accordingly.

Q5: What are practical applications of this calculation?
A: This calculation is useful in architecture, mechanical engineering, graphic design, and any field where rectangular objects need to be inscribed in circular boundaries or where the circular enclosure of rectangular elements needs to be determined.