Circumradius of Rectangle given Area and Obtuse Angle between Diagonals Calculator

Circumradius of Rectangle Formula:

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

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. Every rectangle is cyclic, meaning all its vertices lie on a single circle, and the center of this circle is the intersection point of the diagonals.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( A \) — Area of Rectangle (m²)
\( \angle_{d(Obtuse)} \) — Obtuse Angle between Diagonals (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)

3. Formula Explanation

Details: This formula derives from the geometric properties of rectangles and trigonometric relationships between the area, diagonal angles, and circumradius. The cotangent and cosine functions help relate the area and angle to the circumradius.

4. Using the Calculator

Tips: Enter the area of the rectangle in square meters and the obtuse angle between diagonals in degrees (must be between 90° and 180°). The calculator will compute the circumradius in meters.

5. Frequently Asked Questions (FAQ)

Q1: Why is there an obtuse angle parameter?
A: In a rectangle, the diagonals intersect at two different angles - one acute and one obtuse. The obtuse angle is used in this specific formula derivation.

Q2: What is the relationship between circumradius and diagonal?
A: The circumradius of a rectangle is exactly half the length of its diagonal.

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

Q4: What are the units of measurement?
A: The area should be in square meters and the result (circumradius) will be in meters. You can use any consistent unit system.

Q5: Why does the formula use trigonometric functions?
A: The trigonometric functions (cotangent and cosine) help express the relationship between the area, the angle between diagonals, and the circumradius in a compact mathematical form.