Circumradius of Rectangle Given Breadth and Angle between Diagonal and Length Calculator

Circumradius of Rectangle Formula:

\[ r_c = \frac{1}{2} \times (b \times \cosec(\angle_{dl})) \]

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 rectangles are cyclic quadrilaterals, they can be inscribed in a circle, and the circumradius is half the length of the diagonal.

2. How Does the Calculator Work?

The calculator uses the circumradius formula:

\[ r_c = \frac{1}{2} \times (b \times \cosec(\angle_{dl})) \]

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( b \) — Breadth of Rectangle (m)
\( \angle_{dl} \) — Angle between Diagonal and Length of Rectangle (°)
\( \cosec \) — Cosecant trigonometric function

Explanation: The formula calculates the circumradius using the breadth of the rectangle and the angle between the diagonal and the length, applying trigonometric relationships.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry, engineering, and design applications where circular patterns or circumscribed geometries are involved. It helps determine the minimum circle that can contain a given rectangle.

4. Using the Calculator

Tips: Enter the breadth of the rectangle in meters and the angle between the diagonal and length in degrees. Both values must be positive, and the angle should be between 0° and 90°.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between circumradius and diagonal?
A: The circumradius of a rectangle is exactly half the length of its diagonal.

Q2: Can this formula be used for all rectangles?
A: Yes, this formula applies to all rectangles since all rectangles are cyclic quadrilaterals.

Q3: What happens if the angle is 90 degrees?
A: If the angle is 90 degrees, it means the rectangle is actually a square, and the formula still applies correctly.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact, assuming precise input values and proper trigonometric function implementation.

Q5: What are practical applications of circumradius calculation?
A: Practical applications include architectural design, mechanical engineering, computer graphics, and any field involving geometric computations with rectangles and circles.