Circumradius of Rectangle given Perimeter and Angle between Diagonal and Length Calculator

Formula Used:

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

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:

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

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( P \) — Perimeter of Rectangle (m)
\( \angle_{dl} \) — Angle between Diagonal and Length of Rectangle (rad)

Explanation: The formula calculates the circumradius based on the perimeter of the rectangle and the angle between its diagonal and length, using trigonometric functions to account for the geometric relationships.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry and various engineering applications where circular containment or circumscribed circles around rectangular shapes are involved.

4. Using the Calculator

Tips: Enter the perimeter in meters and the angle in radians. Both values must be positive numbers for valid calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: Can this formula be used for squares?
A: Yes, since a square is a special case of rectangle, this formula applies to squares as well.

Q3: What is the range of valid angle values?
A: The angle between diagonal and length should be between 0 and π/2 radians (0-90 degrees).

Q4: How is the perimeter related to the circumradius?
A: The perimeter provides information about the sum of all sides, which combined with the angle helps determine the individual dimensions and thus the circumradius.

Q5: Are there alternative methods to calculate circumradius?
A: Yes, circumradius can also be calculated using the length and width of the rectangle: \( r_c = \frac{\sqrt{l^2 + w^2}}{2} \)