Volume of a Parallelepiped Calculator with Angles

Parallelepiped Volume Formula:

\[ V = a b c \sqrt{1 + 2 \cos \alpha \cos \beta \cos \gamma - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma} \]

Edge length (a):

Edge length (b):

Edge length (c):

Angle α (between b and c):

degrees

Angle β (between a and c):

degrees

Angle γ (between a and b):

degrees

Volume (V):

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1. What is the Parallelepiped Volume Formula?

The volume of a parallelepiped can be calculated using its three edge lengths and the three angles between them. This formula is derived from the scalar triple product of vectors representing the edges.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V = a b c \sqrt{1 + 2 \cos \alpha \cos \beta \cos \gamma - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma} \]

Where:

\( a, b, c \) — Edge lengths (in meters)
\( \alpha \) — Angle between edges b and c (in degrees)
\( \beta \) — Angle between edges a and c (in degrees)
\( \gamma \) — Angle between edges a and b (in degrees)
\( V \) — Volume (in cubic meters)

Explanation: The formula accounts for the spatial arrangement of the edges through the angles between them.

3. Importance of Volume Calculation

Details: Calculating the volume of a parallelepiped is essential in crystallography, material science, and engineering where objects have oblique angles.

4. Using the Calculator

Tips: Enter all three edge lengths in meters and all three angles in degrees. Angles must be between 0° and 180°.

5. Frequently Asked Questions (FAQ)

Q1: What is a parallelepiped?
A: A parallelepiped is a three-dimensional figure formed by six parallelograms, like a cube but with angles that aren't necessarily 90 degrees.

Q2: What if all angles are 90 degrees?
A: The formula simplifies to V = a × b × c, which is the volume of a rectangular prism.

Q3: Can I use different units?
A: Yes, but all length units must be the same. The volume will be in cubic units of whatever length unit you use.

Q4: What are valid angle ranges?
A: Each angle must be between 0° and 180° for the formula to be valid.

Q5: Why does the formula give zero for some angles?
A: When the angles make the figure flat (coplanar vectors), the volume becomes zero.