Perimeter Of Rectangle Given Circumradius And Obtuse Angle Between Diagonals Calculator

Formula Used:

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

Perimeter of Rectangle:

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1. What is the Perimeter of Rectangle Formula?

The formula calculates the perimeter of a rectangle using its circumradius and the obtuse angle between its diagonals. This provides an alternative method to the standard perimeter calculation when these specific parameters are known.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( P \) — Perimeter of Rectangle (m)
\( r_c \) — Circumradius of Rectangle (m)
\( \angle_{d(Obtuse)} \) — Obtuse Angle between Diagonals (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula derives from trigonometric relationships between the rectangle's diagonals, circumradius, and the angles between them.

3. Importance of Perimeter Calculation

Details: Calculating perimeter is essential in various applications including construction, land surveying, material estimation, and geometric problem solving.

4. Using the Calculator

Tips: Enter circumradius in meters and obtuse angle between diagonals in degrees (must be between 90° and 180°). Both values must be positive.

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. For a rectangle, it's half the length of the diagonal.

Q2: Why is the angle specified as obtuse?
A: In a rectangle, the diagonals create two angles at their intersection - one acute and one obtuse. The formula specifically uses the obtuse angle.

Q3: Can I use this formula for squares?
A: Yes, since a square is a special case of rectangle. For a square, the obtuse angle between diagonals is 135°.

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

Q5: How does this compare to the standard perimeter formula?
A: The standard formula P = 2(length + width) requires knowing side lengths, while this formula uses geometric properties related to the circumscribed circle and diagonal angles.