Length of Rectangle given Area and Acute Angle between Diagonals Calculator

Formula Used:

\[ l = \sqrt{\frac{A}{\tan\left(\frac{\theta}{2}\right)}} \]

Length of Rectangle (l):

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1. What is the Length of Rectangle Formula?

The formula calculates the length of a rectangle when given its area and the acute angle between its diagonals. This geometric relationship helps determine the dimensions of a rectangle based on its total area and diagonal intersection properties.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l = \sqrt{\frac{A}{\tan\left(\frac{\theta}{2}\right)}} \]

Where:

\( l \) — Length of Rectangle (m)
\( A \) — Area of Rectangle (m²)
\( \theta \) — Acute Angle between Diagonals (degrees)

Explanation: The formula derives from the trigonometric relationships between the rectangle's dimensions, area, and the angle formed by its diagonals.

3. Importance of Rectangle Length Calculation

Details: Calculating rectangle length from area and diagonal angle is crucial in geometry, architecture, engineering, and design applications where specific dimensional relationships are required.

4. Using the Calculator

Tips: Enter area in square meters and acute angle in degrees (must be between 0 and 90 degrees). All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: Why is the angle divided by 2 in the formula?
A: The diagonals of a rectangle bisect each other and create congruent triangles. The half-angle appears due to the trigonometric relationships in these right triangles.

Q2: What is the range of valid angle values?
A: The acute angle between diagonals must be between 0 and 90 degrees. A 90-degree angle would indicate a square.

Q3: Can this formula be used for squares?
A: Yes, for a square, the acute angle between diagonals is 90 degrees, and the formula will correctly calculate the side length.

Q4: What if I have the obtuse angle instead of the acute angle?
A: The obtuse angle between diagonals is supplementary to the acute angle (sums to 180 degrees). You can calculate the acute angle by subtracting the obtuse angle from 180 degrees.

Q5: Are there any limitations to this formula?
A: The formula assumes a perfect rectangle and may not account for measurement errors or irregular shapes. The angle must be acute (less than 90 degrees).