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

Formula Used:

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

Circumradius of Rectangle:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Circumradius of Rectangle?

The circumradius of a rectangle is the radius of the circle that passes through all four vertices of the rectangle. 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:

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

Where:

\( r_c \) — Circumradius of Rectangle (m)
\( P \) — Perimeter of Rectangle (m)
\( \angle_{db} \) — Angle between Diagonal and Breadth of Rectangle (radians)
\( \pi \) — Archimedes' constant (approximately 3.14159)

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

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry, engineering, and design applications where circular patterns or constraints involving rectangles are involved.

4. Using the Calculator

Tips: Enter the perimeter of the rectangle in meters and the angle between the diagonal and breadth in degrees. The angle must be between 0° and 90°.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between circumradius and diagonal?
A: The circumradius of a rectangle is exactly half the length of its diagonal.

Q2: Can the circumradius be larger than the rectangle's dimensions?
A: Yes, the circumradius is always equal to half the diagonal length, which is the longest straight line within the rectangle.

Q3: How does the angle affect the circumradius?
A: Different angles between the diagonal and breadth result in different rectangle proportions, which affect both the diagonal length and consequently the circumradius.

Q4: Is this formula applicable to all rectangles?
A: Yes, this formula works for all rectangles regardless of their proportions.

Q5: What are practical applications of circumradius calculation?
A: Useful in architecture, mechanical design, computer graphics, and any field where rectangular shapes need to be inscribed in circles or circular patterns.