Length of Rectangle given Area and Circumradius Calculator

Formula Used:

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

Length of Rectangle (l):

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1. What is the Length of Rectangle given Area and Circumradius Formula?

The formula calculates the length of a rectangle when its area and circumradius (radius of the circumscribed circle) are known. It provides a mathematical relationship between these geometric properties of a rectangle.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( l \) — Length of Rectangle (m)
\( r_c \) — Circumradius of Rectangle (m)
\( A \) — Area of Rectangle (m²)

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

3. Importance of Length Calculation

Details: Calculating the length of a rectangle from its area and circumradius is important in various geometric applications, architectural design, and engineering calculations where these specific parameters are known.

4. Using the Calculator

Tips: Enter the circumradius in meters, area in square meters. All values must be valid positive numbers that satisfy the geometric constraints (4*r_c⁴ must be greater than A² for real solutions).

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: When does this formula give real solutions?
A: The formula gives real solutions when 4*r_c⁴ ≥ A², which is always true for valid rectangles since the maximum area for a given circumradius occurs in a square.

Q3: Can this formula be used for squares?
A: Yes, for a square (where length = width), the formula simplifies appropriately since both area and circumradius have specific relationships with the side length.

Q4: What are the units for the inputs and outputs?
A: The calculator uses meters for length and circumradius, and square meters for area. Ensure consistent units for accurate results.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of rectangles, though practical accuracy depends on the precision of input measurements.