Length of Rectangle Given Diameter of Circumcircle and Acute Angle between Diagonals Calculator

Formula Used:

\[ l = D_c \times \cos\left(\frac{\theta_{acute}}{2}\right) \]

Length of Rectangle (l):

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1. What is the Length of Rectangle Given Diameter of Circumcircle and Acute Angle between Diagonals?

This calculation determines the length of a rectangle using the diameter of its circumcircle and the acute angle between its diagonals. The circumcircle is the circle that passes through all four vertices of the rectangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ l = D_c \times \cos\left(\frac{\theta_{acute}}{2}\right) \]

Where:

\( l \) — Length of Rectangle (m)
\( D_c \) — Diameter of Circumcircle of Rectangle (m)
\( \theta_{acute} \) — Acute Angle between Diagonals of Rectangle (°)

Explanation: The formula derives from trigonometric relationships in a rectangle, where the diagonal equals the diameter of the circumcircle, and the angle between diagonals relates to the rectangle's aspect ratio.

3. Importance of Length Calculation

Details: Accurate length calculation is essential in geometry, engineering, and design applications where rectangular shapes are involved, particularly when working with circumscribed circles.

4. Using the Calculator

Tips: Enter the diameter of the circumcircle in meters and the acute angle between diagonals in degrees (must be between 0° and 90°). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is a circumcircle of a rectangle?
A: A circumcircle is a circle that passes through all four vertices of a rectangle. For any rectangle, the circumcircle's center is at the intersection of the diagonals.

Q2: Why is the angle divided by 2 in the formula?
A: The division by 2 accounts for the trigonometric relationship where the length relates to half the angle between diagonals in the right triangle formed.

Q3: Can this formula be used for any rectangle?
A: Yes, this formula applies to all rectangles, as the relationship between the circumcircle diameter, diagonal angle, and side lengths is consistent.

Q4: What is the range of valid angles?
A: The acute angle between diagonals must be between 0° and 90°. At 0°, the rectangle would be degenerate, and at 90°, it would be a square.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect rectangles. Accuracy depends on the precision of input measurements.