Length of Rectangle given Circumradius and Acute Angle between Diagonals Calculator

Formula Used:

\[ l = 2 \times r_c \times \cos\left(\frac{\angle d_{Acute}}{2}\right) \]

Length of Rectangle:

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1. What is the Length of Rectangle given Circumradius and Acute Angle between Diagonals Formula?

This formula calculates the length of a rectangle when you know its circumradius (radius of the circumscribed circle) and the acute angle between its diagonals. It's derived from geometric relationships in rectangles and trigonometric functions.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l = 2 \times r_c \times \cos\left(\frac{\angle d_{Acute}}{2}\right) \]

Where:

\( l \) — Length of Rectangle (m)
\( r_c \) — Circumradius of Rectangle (m)
\( \angle d_{Acute} \) — Acute Angle between Diagonals of Rectangle (radians)

Explanation: The formula uses the cosine function to relate the circumradius and the acute angle between diagonals to determine the length of the rectangle.

3. Importance of Length Calculation

Details: Calculating the length of a rectangle using circumradius and diagonal angle is important in geometry problems, architectural design, and engineering applications where these specific parameters are known.

4. Using the Calculator

Tips: Enter circumradius in meters and acute angle between diagonals in degrees. The angle must be between 0° and 90° (exclusive). All values must be positive.

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's half the length of the diagonal.

Q2: Why is the angle divided by 2 in the formula?
A: The division by 2 comes from the trigonometric relationship between the diagonal, the sides of the rectangle, and the angles formed.

Q3: Can this formula be used for squares?
A: Yes, since a square is a special case of a rectangle, this formula applies to squares as well.

Q4: What if I know the obtuse angle between diagonals?
A: The acute and obtuse angles between diagonals are supplementary (add up to 180°). You can calculate the acute angle by subtracting the obtuse angle from 180°.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the geometric properties of rectangles, assuming precise input values.