Diameter of Circumcircle of Rectangle given Perimeter and Obtuse Angle between Diagonals Calculator

Formula Used:

\[ D_c = \frac{P}{2} \times \frac{1}{\sqrt{1 + \sin\left(2 \times \frac{\pi - \angle_{d(Obtuse)}}{2}\right)}} \]

Diameter of Circumcircle of Rectangle:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Diameter of Circumcircle of Rectangle?

The diameter of the circumcircle of a rectangle is the diameter of the circle that passes through all four vertices of the rectangle. For any rectangle, the circumcircle's center is at the intersection point of the diagonals, and its diameter equals the length of the diagonal of the rectangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ D_c = \frac{P}{2} \times \frac{1}{\sqrt{1 + \sin\left(2 \times \frac{\pi - \angle_{d(Obtuse)}}{2}\right)}} \]

Where:

\( D_c \) — Diameter of Circumcircle of Rectangle (m)
\( P \) — Perimeter of Rectangle (m)
\( \angle_{d(Obtuse)} \) — Obtuse Angle between Diagonals of Rectangle (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: This formula derives the diameter of the circumcircle from the perimeter and the obtuse angle between the diagonals of the rectangle, utilizing trigonometric relationships.

3. Importance of Circumcircle Diameter Calculation

Details: Calculating the diameter of the circumcircle is important in geometry and various practical applications, such as construction and design, where understanding the spatial properties of rectangles and their circumscribed circles is necessary.

4. Using the Calculator

Tips: Enter the perimeter of the rectangle in meters and the obtuse angle between the diagonals in degrees. The angle must be between 90 and 180 degrees. All values must be valid (perimeter > 0, angle between 90-180).

5. Frequently Asked Questions (FAQ)

Q1: Why is the angle between diagonals important?
A: The angle between diagonals affects the shape of the rectangle and consequently the diameter of its circumcircle. Different angles result in different aspect ratios of the rectangle.

Q2: What is the relationship between the diagonal and the circumcircle diameter?
A: For any rectangle, the diagonal length equals the diameter of its circumcircle. This calculator finds that diameter using perimeter and the obtuse angle between diagonals.

Q3: Can this formula be used for squares?
A: Yes, a square is a special case of a rectangle where all angles between diagonals are 90 degrees (which is not obtuse). For squares, the formula simplifies as the angle parameter becomes 90 degrees.

Q4: What are the units of measurement?
A: The perimeter should be entered in meters, and the result (diameter) will be in meters. The angle is in degrees but is converted to radians for calculation.

Q5: Are there limitations to this formula?
A: The formula is mathematically precise for all rectangles. However, numerical precision might be affected by extreme values due to floating-point arithmetic limitations in computing.