Circumradius of Rectangle given Breadth and Obtuse Angle between Diagonals Calculator

Formula Used:

\[ r_c = \frac{1}{2} \times \left( b \times \csc\left(\frac{\pi - \angle_{d(Obtuse)}}{2}\right) \right) \]

Circumradius of Rectangle:

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1. What is Circumradius of Rectangle?

The circumradius of a rectangle is the radius of the circle that passes through all four vertices of the rectangle. Since all vertices of a rectangle lie on a circle, every rectangle is cyclic.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{1}{2} \times \left( b \times \csc\left(\frac{\pi - \angle_{d(Obtuse)}}{2}\right) \right) \]

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( b \) — Breadth of Rectangle (m)
\( \angle_{d(Obtuse)} \) — Obtuse Angle between Diagonals of Rectangle (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \csc \) — Cosecant trigonometric function

Explanation: The formula calculates the circumradius using the breadth of the rectangle and the obtuse angle between its diagonals through trigonometric relationships.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry problems, architectural design, and engineering applications where circular properties of rectangular shapes need to be determined.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: Why is the circumradius important for a rectangle?
A: The circumradius helps determine the smallest circle that can circumscribe the rectangle, which is useful in various geometric and design applications.

Q2: What is the relationship between diagonals and circumradius?
A: In a rectangle, the circumradius is exactly half the length of the diagonal, as the diagonals are equal and serve as diameters of the circumcircle.

Q3: Can this formula be used for all rectangles?
A: Yes, this formula applies to all rectangles since every rectangle is cyclic and has a circumcircle passing through all four vertices.

Q4: What if I know the length instead of breadth?
A: You would need to use a different formula that incorporates the length of the rectangle instead of the breadth.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the input values, assuming precise trigonometric function computations.