Perimeter Of Rectangle Given Diameter Of Circumcircle And Obtuse Angle Between Diagonals Calculator

Formula Used:

\[ P = 2 \times D_c \times \sqrt{1 + (2 \times \sin\left(\frac{\pi - \angle_d(Obtuse)}{2}\right) \times \cos\left(\frac{\pi - \angle_d(Obtuse)}{2}\right))} \]

Perimeter of Rectangle (P):

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1. What Is The Perimeter Of Rectangle Given Diameter Of Circumcircle And Obtuse Angle Between Diagonals?

This calculator determines the perimeter of a rectangle when given the diameter of its circumcircle and the obtuse angle between its diagonals. The circumcircle is the circle that passes through all four vertices of the rectangle.

2. How Does The Calculator Work?

The calculator uses the formula:

\[ P = 2 \times D_c \times \sqrt{1 + (2 \times \sin\left(\frac{\pi - \angle_d(Obtuse)}{2}\right) \times \cos\left(\frac{\pi - \angle_d(Obtuse)}{2}\right))} \]

Where:

\( P \) — Perimeter of Rectangle (m)
\( D_c \) — Diameter of Circumcircle of Rectangle (m)
\( \angle_d(Obtuse) \) — Obtuse Angle between Diagonals of Rectangle (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula relates the perimeter of a rectangle to the diameter of its circumscribed circle and the angle between its diagonals using trigonometric functions.

3. Importance Of Perimeter Calculation

Details: Calculating the perimeter of a rectangle is fundamental in geometry and has practical applications in construction, design, and various engineering fields where rectangular shapes are involved.

4. Using The Calculator

Tips: Enter the diameter of the circumcircle in meters and the obtuse angle between diagonals in degrees (between 90° and 180°). All values must be valid positive numbers.

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 of the rectangle.

Q2: Why is the angle between diagonals important?
A: The angle between diagonals helps determine the aspect ratio of the rectangle and is crucial for calculating various geometric properties including perimeter and area.

Q3: What is the range of obtuse angles between diagonals?
A: For a rectangle, the obtuse angle between diagonals ranges from 90° to 180°, with 90° representing a square and larger angles representing more elongated rectangles.

Q4: Can this formula be used for squares?
A: Yes, for a square (where all sides are equal), the obtuse angle between diagonals is 90°, and the formula simplifies accordingly.

Q5: What are the units of measurement?
A: The diameter should be entered in meters, and the result will be in meters. The angle is entered in degrees but converted to radians for calculation.