Breadth of Rectangle given Area and Circumradius Calculator

Formula Used:

\[ Breadth = \sqrt{(2 \times Circumradius^2) - \sqrt{(4 \times Circumradius^4) - Area^2}} \]

Breadth of Rectangle:

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1. What is the Breadth of Rectangle Formula?

The formula calculates the breadth of a rectangle when given its area and circumradius. The circumradius is the radius of the circle that passes through all four vertices of the rectangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Breadth = \sqrt{(2 \times r_c^2) - \sqrt{(4 \times r_c^4) - A^2}} \]

Where:

\( A \) — Area of the rectangle (m²)
\( r_c \) — Circumradius of the rectangle (m)

Explanation: This formula derives from the geometric relationships between a rectangle's dimensions, its area, and the circumradius of its circumscribed circle.

3. Importance of Rectangle Dimensions

Details: Calculating precise dimensions of rectangles is essential in various fields including architecture, engineering, and design where accurate measurements are critical.

4. Using the Calculator

Tips: Enter the area in square meters and circumradius in meters. Both values must be positive numbers. The calculator will compute the breadth of the rectangle.

5. Frequently Asked Questions (FAQ)

Q1: What is the circumradius of a rectangle?
A: The circumradius is the radius of the circle that passes through all four vertices of the rectangle. For a rectangle, it equals half the length of the diagonal.

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

Q3: What are the units of measurement?
A: The calculator uses meters for length measurements and square meters for area. Ensure consistent units for accurate results.

Q4: Are there limitations to this formula?
A: The formula requires that the given area and circumradius values form a valid rectangle (4*r_c⁴ must be greater than A²).

Q5: How is this formula derived?
A: The formula is derived from the Pythagorean theorem and the relationships between a rectangle's dimensions, area, and diagonal length.