Area of Rectangle given Circumradius and Acute Angle between Diagonals Calculator

Formula Used:

\[ A = 4 \times r_c^2 \times \cos\left(\frac{\angle_d(acute)}{2}\right) \times \sin\left(\frac{\angle_d(acute)}{2}\right) \]

Area of Rectangle:

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1. What is the Area of Rectangle given Circumradius and Acute Angle between Diagonals Formula?

This formula calculates the area of a rectangle using its circumradius (radius of the circumscribed circle) and the acute angle between its diagonals. It provides an alternative method to calculate area without knowing the side lengths directly.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A = 4 \times r_c^2 \times \cos\left(\frac{\angle_d(acute)}{2}\right) \times \sin\left(\frac{\angle_d(acute)}{2}\right) \]

Where:

\( A \) — Area of Rectangle (m²)
\( r_c \) — Circumradius of Rectangle (m)
\( \angle_d(acute) \) — Acute Angle between Diagonals of Rectangle (radians)

Explanation: The formula utilizes trigonometric functions to relate the circumradius and diagonal angle to the area of the rectangle.

3. Importance of Area Calculation

Details: Calculating area using circumradius and diagonal angle is useful in geometry problems where these parameters are known instead of side lengths, particularly in circumscribed geometric configurations.

4. Using the Calculator

Tips: Enter circumradius in meters and acute angle between diagonals in degrees. Both values must be positive, with angle between 0-90 degrees.

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 equals half the length of the diagonal.

Q2: Why use this formula instead of length × width?
A: This formula is useful when you know the circumradius and diagonal angle but not the side lengths directly.

Q3: What is the range of valid angles for this formula?
A: The acute angle between diagonals must be between 0 and 90 degrees (0° < angle ≤ 90°).

Q4: How is this formula derived?
A: The formula is derived from trigonometric relationships between the diagonal length, circumradius, and the angle between diagonals in a rectangle.

Q5: Can this formula be used for other quadrilaterals?
A: No, this specific formula applies only to rectangles where diagonals are equal and bisect each other.