Obtuse Angle between Diagonals of Rectangle given Length and Circumradius Calculator

Formula Used:

\[ \text{Obtuse Angle between Diagonals of Rectangle} = 2 \times \arcsin\left(\frac{\text{Length of Rectangle}}{2 \times \text{Circumradius of Rectangle}}\right) \]

Obtuse Angle between Diagonals of Rectangle:

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1. What is Obtuse Angle between Diagonals of Rectangle?

The Obtuse Angle between Diagonals of Rectangle is the angle greater than 90 degrees formed by the intersection of the two diagonals in a rectangle. In any rectangle, the diagonals are equal in length and bisect each other, creating two acute and two obtuse angles at their intersection point.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Obtuse Angle} = 2 \times \arcsin\left(\frac{\text{Length}}{2 \times \text{Circumradius}}\right) \]

Where:

Length — Length of the Rectangle (m)
Circumradius — Radius of the circumscribed circle containing all vertices of the rectangle (m)

Explanation: This formula derives from the geometric relationship between the rectangle's dimensions and its circumradius, using trigonometric functions to calculate the angle between diagonals.

3. Importance of Angle Calculation

Details: Calculating the angle between diagonals is important in geometry, engineering, and design applications where precise angular measurements are required for structural analysis and spatial planning.

4. Using the Calculator

Tips: Enter the length of the rectangle and its circumradius in meters. Both values must be positive numbers. The calculator will compute the obtuse angle between the diagonals in degrees.

5. Frequently Asked Questions (FAQ)

Q1: Why is there an obtuse angle between diagonals?
A: In a rectangle, diagonals intersect and create four angles - two acute and two obtuse. The obtuse angles are always greater than 90 degrees due to the rectangular shape's properties.

Q2: How does circumradius relate to rectangle dimensions?
A: The circumradius of a rectangle is half the length of its diagonal, connecting the relationship between the rectangle's dimensions and the circle that circumscribes it.

Q3: Can this formula be used for squares?
A: Yes, since a square is a special case of rectangle, the formula applies. In a square, the obtuse angle between diagonals would be 135 degrees.

Q4: What is the range of possible obtuse angles?
A: The obtuse angle between diagonals of a rectangle ranges from 90 degrees (approaching a degenerate case) to 180 degrees (theoretical maximum), though practical rectangles typically have angles between 90 and 180 degrees.

Q5: Are both obtuse angles equal in a rectangle?
A: Yes, in any rectangle, the two obtuse angles formed by the intersecting diagonals are always equal to each other, as are the two acute angles.