Circumradius of Rectangle Given Perimeter and Acute Angle Between Diagonals Calculator

Formula Used:

\[ r_c = \frac{1}{2} \times \left( \frac{P}{2} \times \frac{1}{\sqrt{1 + \sin\left(2 \times \frac{\angle_{d(acute)}}{2}\right)}} \right) \]

Circumradius of Rectangle:

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1. What is Circumradius of Rectangle?

The circumradius of a rectangle is the radius of the circle which contains the rectangle with all the vertices of the rectangle lying on the circle. 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:

\[ r_c = \frac{1}{2} \times \left( \frac{P}{2} \times \frac{1}{\sqrt{1 + \sin\left(2 \times \frac{\angle_{d(acute)}}{2}\right)}} \right) \]

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( P \) — Perimeter of Rectangle (m)
\( \angle_{d(acute)} \) — Acute Angle between Diagonals of Rectangle (degrees)

Explanation: This formula calculates the circumradius based on the perimeter and the acute angle between the diagonals of the rectangle, using trigonometric relationships.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry and various engineering applications where the relationship between a rectangle and its circumscribed circle needs to be determined.

4. Using the Calculator

Tips: Enter the perimeter of the rectangle in meters and the acute angle between diagonals in degrees. The angle must be between 0 and 90 degrees, and the perimeter 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, since a square is a special type of rectangle, this formula works for squares as well.

Q3: What is the range of possible values for the acute angle?
A: The acute angle between diagonals of a rectangle ranges from 0° to 90°, with 90° occurring only in a square.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact based on the input values, assuming perfect geometric conditions.

Q5: What are the practical applications of circumradius calculation?
A: This calculation is used in various fields including architecture, engineering, computer graphics, and geometric design.