Perimeter of Rectangle given Circumradius and Angle between Diagonal and Breadth Calculator

Formula Used:

\[ P = 4 \times r_c \times \sqrt{1 + (2 \times \sin((\pi/2) - \angle_{db}) \times \cos((\pi/2) - \angle_{db}))} \]

Perimeter of Rectangle:

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1. What is the Perimeter of Rectangle given Circumradius and Angle between Diagonal and Breadth?

This formula calculates the perimeter of a rectangle when you know its circumradius (radius of the circumscribed circle) and the angle between the diagonal and the breadth (width) of the rectangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ P = 4 \times r_c \times \sqrt{1 + (2 \times \sin((\pi/2) - \angle_{db}) \times \cos((\pi/2) - \angle_{db}))} \]

Where:

\( P \) — Perimeter of Rectangle (m)
\( r_c \) — Circumradius of Rectangle (m)
\( \angle_{db} \) — Angle between Diagonal and Breadth of Rectangle (rad)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula derives from trigonometric relationships between the rectangle's dimensions, its circumradius, and the angle between its diagonal and breadth.

3. Importance of Perimeter Calculation

Details: Calculating perimeter is essential in various fields including construction, manufacturing, and geometry problems. It helps determine the total boundary length of rectangular objects.

4. Using the Calculator

Tips: Enter circumradius in meters and angle in radians. The angle should be between 0 and π/2 radians (0-90 degrees). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is circumradius of a rectangle?
A: Circumradius is the radius of the circle that passes through all four vertices of the rectangle. For a rectangle, it's half the length of the diagonal.

Q2: Why use this specific formula?
A: This formula provides a direct way to calculate perimeter when you know the circumradius and the angle between diagonal and breadth, without needing to find individual side lengths first.

Q3: What are the valid angle ranges?
A: The angle between diagonal and breadth must be between 0 and π/2 radians (0-90 degrees). At 0 radians, the rectangle becomes a line, and at π/2 radians, it becomes a square.

Q4: Can I use degrees instead of radians?
A: The calculator requires radians. To convert degrees to radians, multiply by π/180 (approximately 0.0174533).

Q5: What are practical applications of this calculation?
A: Useful in engineering design, architectural planning, and any situation where you need to determine the boundary length of a rectangular space with given circumscribed circle properties.