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

Formula Used:

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

Circumradius of Rectangle:

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1. What is Circumradius of Rectangle?

The circumradius of a rectangle is the radius of the circle which contains the rectangle with all the vertices of the rectangle lying on the circle. It represents the distance from the center of the rectangle to any of its vertices.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( b \) — Breadth of Rectangle (m)
\( \angle_{db} \) — Angle between Diagonal and Breadth of Rectangle (rad)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \csc \) — Cosecant trigonometric function

Explanation: The formula calculates the circumradius using the breadth of the rectangle and the angle between the diagonal and breadth, applying trigonometric relationships in a right-angled triangle.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry and various engineering applications where circular patterns or circumscribed circles around rectangular shapes are involved, such as in mechanical design and architectural planning.

4. Using the Calculator

Tips: Enter the breadth of the rectangle in meters and the angle between the diagonal and breadth in radians. Both values must be positive numbers.

5. Frequently Asked Questions (FAQ)

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

Q2: How is the angle between diagonal and breadth measured?
A: The angle is measured in radians between the diagonal and the shorter side (breadth) of the rectangle.

Q3: Can this calculator handle degrees instead of radians?
A: The calculator requires angle input in radians. Convert degrees to radians by multiplying by π/180.

Q4: What is the range of valid angle values?
A: The angle should be between 0 and π/2 radians (0° to 90°) for a valid rectangle.

Q5: How accurate is the calculation?
A: The calculation uses precise trigonometric functions and provides results accurate to 6 decimal places.