Area of Rectangle Given Diameter of Circumcircle and Acute Angle between Diagonals Calculator

Area of Rectangle Formula:

\[ A = D_c^2 \times \sin\left(\frac{\angle d_{\text{(Acute)}}}{2}\right) \times \cos\left(\frac{\angle d_{\text{(Acute)}}}{2}\right) \]

Area of Rectangle:

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1. What is the Area of Rectangle Given Diameter of Circumcircle and Acute Angle between Diagonals?

This formula calculates the area of a rectangle using the diameter of its circumcircle and the acute angle between its diagonals. It provides an alternative method to calculate area when these specific parameters are known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A = D_c^2 \times \sin\left(\frac{\angle d_{\text{(Acute)}}}{2}\right) \times \cos\left(\frac{\angle d_{\text{(Acute)}}}{2}\right) \]

Where:

\( A \) — Area of Rectangle (m²)
\( D_c \) — Diameter of Circumcircle of Rectangle (m)
\( \angle d_{\text{(Acute)}} \) — Acute Angle between Diagonals of Rectangle (degrees)

Explanation: The formula utilizes trigonometric relationships between the rectangle's diagonals, the circumcircle diameter, and the angle between diagonals to determine the area.

3. Importance of Area Calculation

Details: Calculating the area of a rectangle is fundamental in geometry and has practical applications in various fields including construction, design, and engineering.

4. Using the Calculator

Tips: Enter the diameter of the circumcircle in meters and the acute angle between diagonals in degrees. The angle must be between 0 and 90 degrees.

5. Frequently Asked Questions (FAQ)

Q1: What is a circumcircle of a rectangle?
A: A circumcircle is a circle that passes through all four vertices of the rectangle. For a rectangle, the diameter of the circumcircle equals the length of the diagonal.

Q2: Why use this formula instead of length × width?
A: This formula is useful when you know the circumcircle diameter and the angle between diagonals but not the side lengths of the rectangle.

Q3: What is the range of valid angles for this formula?
A: The acute angle between diagonals must be between 0 and 90 degrees. For a rectangle, this angle is always acute.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact, assuming precise input values. The calculator provides results with 6 decimal places for accuracy.

Q5: Can this formula be used for squares?
A: Yes, for a square (which is a special case of rectangle), the angle between diagonals is 90 degrees, and the formula simplifies accordingly.