Circumradius of Rectangle Given Area and Angle Between Diagonal and Breadth Calculator

Formula Used:

\[ r_c = \frac{1}{2} \times \frac{\sqrt{A \times \cot(\frac{\pi}{2} - \angle_{db})}}{\cos(\frac{\pi}{2} - \angle_{db})} \]

Circumradius of Rectangle (r_c):

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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. For any rectangle, the circumradius is equal to half the length of the diagonal.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_c = \frac{1}{2} \times \frac{\sqrt{A \times \cot(\frac{\pi}{2} - \angle_{db})}}{\cos(\frac{\pi}{2} - \angle_{db})} \]

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( A \) — Area of Rectangle (m²)
\( \angle_{db} \) — Angle between Diagonal and Breadth (degrees)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: This formula calculates the circumradius using the area of the rectangle and the angle between its diagonal and breadth, utilizing trigonometric relationships.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry, engineering, and design applications where circular patterns or circumscribed circles around rectangular objects are involved.

4. Using the Calculator

Tips: Enter the area of the rectangle in square meters and the angle between the diagonal and breadth in degrees (0-90°). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between circumradius and diagonal?
A: For any rectangle, the circumradius is exactly half the length of the diagonal.

Q2: Can this formula be used for squares?
A: Yes, squares are special cases of rectangles, so the formula applies to squares as well.

Q3: What is the range of possible circumradius values?
A: The circumradius can be any positive value, depending on the size of the rectangle.

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

Q5: What if the angle is 45 degrees?
A: When the angle is 45 degrees, the rectangle becomes a square, and the formula simplifies accordingly.