Breadth of Rectangle given Circumradius and Acute Angle between Diagonals Calculator

Formula Used:

\[ b = 2 \times r_c \times \sin(\frac{\angle_d(Acute)}{2}) \]

Breadth of Rectangle:

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1. What is the Breadth of Rectangle Formula?

The formula calculates the breadth of a rectangle using its circumradius and the acute angle between its diagonals. This geometric relationship helps determine one dimension of the rectangle when the circumradius and angle between diagonals are known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ b = 2 \times r_c \times \sin(\frac{\angle_d(Acute)}{2}) \]

Where:

\( b \) — Breadth of Rectangle (m)
\( r_c \) — Circumradius of Rectangle (m)
\( \angle_d(Acute) \) — Acute Angle between Diagonals of Rectangle (radians)

Explanation: The formula derives from trigonometric relationships in a rectangle, where the circumradius connects to the geometric properties through the sine function of half the acute angle between diagonals.

3. Importance of Breadth Calculation

Details: Calculating the breadth is essential in geometry for determining rectangle dimensions, architectural design, engineering applications, and various mathematical problems involving rectangular shapes.

4. Using the Calculator

Tips: Enter circumradius in meters, acute angle in degrees (between 0 and 90). All values must be valid (circumradius > 0, angle between 0-90 degrees).

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, equal to half the length of the diagonal.

Q2: Why use the sine function in this formula?
A: The sine function relates the angle between diagonals to the ratio of the breadth to the circumradius through trigonometric identities in the right triangles formed by the diagonals.

Q3: Can this formula be used for any rectangle?
A: Yes, this formula applies to all rectangles as it's derived from fundamental geometric properties of rectangles.

Q4: What if the angle is obtuse between diagonals?
A: Rectangles always have one acute and one obtuse angle between diagonals. This formula specifically uses the acute angle measurement.

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