Circumradius of Rectangle given Perimeter and Obtuse Angle between Diagonals Calculator

Circumradius of Rectangle Formula:

\[ r_c = \frac{1}{2} \times \left( \frac{P}{2} \times \frac{1}{\sqrt{1 + \sin\left(2 \times \left( \frac{\pi - \angle d_{\text{(Obtuse)}}}{2} \right)\right)}} \right) \]

Circumradius of Rectangle:

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1. What is Circumradius of Rectangle?

The circumradius of a rectangle is the radius of the circle 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 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:

\[ r_c = \frac{1}{2} \times \left( \frac{P}{2} \times \frac{1}{\sqrt{1 + \sin\left(2 \times \left( \frac{\pi - \angle d_{\text{(Obtuse)}}}{2} \right)\right)}} \right) \]

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( P \) — Perimeter of Rectangle (m)
\( \angle d_{\text{(Obtuse)}} \) — Obtuse Angle between Diagonals (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)

3. Formula Explanation

Details: This formula calculates the circumradius of a rectangle using its perimeter and the obtuse angle between its diagonals. The formula incorporates trigonometric functions to account for the geometric relationships between the rectangle's dimensions and its circumscribed circle.

4. Using the Calculator

Tips: Enter the perimeter of the rectangle in meters and the obtuse angle between diagonals in degrees (must be between 90° and 180°). The calculator will automatically convert the angle to radians and compute the circumradius.

5. Frequently Asked Questions (FAQ)

Q1: Why is there an obtuse angle between diagonals in a rectangle?
A: In any rectangle, the diagonals intersect and create two pairs of angles - one acute and one obtuse. The obtuse angle is always greater than 90 degrees.

Q2: What is the relationship between perimeter and circumradius?
A: The perimeter and circumradius are related through the rectangle's side lengths and the trigonometric properties of its diagonals.

Q3: Can this formula be used for squares?
A: Yes, a square is a special case of a rectangle where all sides are equal and the angle between diagonals is 90 degrees.

Q4: What are the units of measurement?
A: The perimeter should be entered in meters, and the result will be in meters. The angle should be entered in degrees.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise based on the input values, using the exact trigonometric relationships defined by the formula.