Angle Beta of Parallelepiped Calculator

Angle Beta of Parallelepiped Formula:

\[ \angle\beta = \sin^{-1}\left(\frac{TSA - (2 \times S_a \times S_b \times \sin(\angle\gamma)) - (2 \times S_b \times S_c \times \sin(\angle\alpha))}{2 \times S_a \times S_c}\right) \]

Total Surface Area of Parallelepiped:

m²

Side A of Parallelepiped:

Side B of Parallelepiped:

Side C of Parallelepiped:

Angle Alpha of Parallelepiped:

rad

Angle Gamma of Parallelepiped:

rad

Angle Beta of Parallelepiped:

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1. What is Angle Beta of Parallelepiped?

Angle Beta of Parallelepiped is the angle formed by side A and side C at any of the two sharp tips of the Parallelepiped. It is one of the three important angles that define the shape and orientation of a parallelepiped in 3D space.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ \angle\beta = \sin^{-1}\left(\frac{TSA - (2 \times S_a \times S_b \times \sin(\angle\gamma)) - (2 \times S_b \times S_c \times \sin(\angle\alpha))}{2 \times S_a \times S_c}\right) \]

Where:

\( TSA \) — Total Surface Area of Parallelepiped (m²)
\( S_a \) — Side A of Parallelepiped (m)
\( S_b \) — Side B of Parallelepiped (m)
\( S_c \) — Side C of Parallelepiped (m)
\( \angle\alpha \) — Angle Alpha of Parallelepiped (rad)
\( \angle\gamma \) — Angle Gamma of Parallelepiped (rad)

Explanation: This formula calculates angle beta based on the total surface area and the other known parameters of the parallelepiped, using trigonometric relationships.

3. Importance of Angle Beta Calculation

Details: Calculating angle beta is essential for understanding the complete geometric configuration of a parallelepiped. It helps in determining the spatial orientation, volume calculations, and various applications in solid geometry, physics, and engineering.

4. Using the Calculator

Tips: Enter all values in the specified units. Ensure that angles are provided in radians and all length measurements are positive. The calculator will compute angle beta using the derived formula.

5. Frequently Asked Questions (FAQ)

Q1: What is a parallelepiped?
A: A parallelepiped is a three-dimensional figure formed by six parallelograms. It's the 3D equivalent of a parallelogram.

Q2: Why are there three different angles in a parallelepiped?
A: The three angles (alpha, beta, gamma) define the orientation of the three pairs of parallel faces and are essential for completely specifying the shape.

Q3: What units should I use for the angles?
A: Angles should be entered in radians. If you have degrees, convert them to radians first (radians = degrees × π/180).

Q4: Can this calculator handle negative values?
A: No, all input values must be positive as they represent physical measurements of length, area, and angles.

Q5: What if I get an error or unexpected result?
A: Check that all input values are valid and consistent. The formula requires that the expression inside the inverse sine function falls between -1 and 1.