Area of Rectangle given Circumradius and Obtuse Angle between Diagonals Calculator

Formula Used:

\[ A = 4 \times r_c^2 \times \cos\left(\frac{\angle_{d(obtuse)}}{2}\right) \times \sin\left(\frac{\angle_{d(obtuse)}}{2}\right) \]

Area of Rectangle:

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1. What is the Area of Rectangle Formula?

The formula calculates the area of a rectangle using its circumradius and the obtuse angle between its diagonals. This provides an alternative method to the standard length × width calculation.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A = 4 \times r_c^2 \times \cos\left(\frac{\angle_{d(obtuse)}}{2}\right) \times \sin\left(\frac{\angle_{d(obtuse)}}{2}\right) \]

Where:

\( A \) — Area of the rectangle
\( r_c \) — Circumradius of the rectangle
\( \angle_{d(obtuse)} \) — Obtuse angle between the diagonals (90° to 180°)

Explanation: The formula utilizes trigonometric relationships between the circumradius, the angle between diagonals, and the resulting area of the rectangle.

3. Importance of Area Calculation

Details: Calculating area using circumradius and diagonal angles is useful in geometric problems where traditional dimensions are not available, particularly in coordinate geometry and architectural design.

4. Using the Calculator

Tips: Enter the circumradius in meters and the obtuse angle between diagonals in degrees (between 90 and 180). Both values must be positive, with the angle within the specified range.

5. Frequently Asked Questions (FAQ)

Q1: What is the circumradius of a rectangle?
A: The circumradius is the radius of the circle that passes through all four vertices of the rectangle. It's equal to half the length of the diagonal.

Q2: Why is the angle between diagonals important?
A: The diagonals of a rectangle intersect at equal angles, and the obtuse angle helps determine the rectangle's proportions and area through trigonometric relationships.

Q3: Can this formula be used for squares?
A: Yes, for a square (which is a special rectangle), the angle between diagonals is 90°, and the formula simplifies accordingly.

Q4: What are the limitations of this formula?
A: The formula assumes a perfect rectangle and requires accurate measurement of the circumradius and the angle between diagonals.

Q5: How does this relate to the standard area formula?
A: This formula provides an alternative approach using geometric properties rather than side lengths, offering flexibility in different problem contexts.