Angle Alpha of Parallelepiped Calculator

Angle Alpha of Parallelepiped Formula:

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

Total Surface Area of Parallelepiped:

m²

Side A of Parallelepiped:

Side B of Parallelepiped:

Side C of Parallelepiped:

Angle Gamma of Parallelepiped:

rad

Angle Beta of Parallelepiped:

rad

Angle Alpha of Parallelepiped:

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1. What is the Angle Alpha of Parallelepiped?

Angle Alpha of Parallelepiped is the angle formed by side B 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 Angle Alpha formula:

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

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)
\( \angle\gamma \) — Angle Gamma of Parallelepiped (rad)
\( \angle\beta \) — Angle Beta of Parallelepiped (rad)

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

3. Importance of Angle Alpha Calculation

Details: Calculating angle alpha is crucial for understanding the complete geometric properties of a parallelepiped, determining its volume, and analyzing its spatial orientation in 3D applications.

4. Using the Calculator

Tips: Enter all values in the specified units (meters for lengths, square meters for area, and radians for angles). All values must be positive and valid for the calculation to work properly.

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 describing the shape's geometry.

Q3: What are the valid ranges for the angles?
A: All three angles in a parallelepiped must be between 0 and π radians (0° and 180°), and their sum must satisfy certain geometric constraints.

Q4: Can this calculator handle degrees instead of radians?
A: The calculator requires angles in radians. To convert from degrees to radians, multiply degrees by π/180.

Q5: What if I get an error or undefined result?
A: This typically occurs when the input values don't form a valid parallelepiped or when the expression inside the inverse sine function falls outside the range [-1, 1].