Parallelepiped Volume Calculator

Calculation Method:

Vector \( \vec{a} \) (x, y, z):

Vector \( \vec{b} \) (x, y, z):

Vector \( \vec{c} \) (x, y, z):

Side \( a \) (cm):

Side \( b \) (cm):

Side \( c \) (cm):

Angle \( \alpha \) (degrees, between \( b \) and \( c \)):

Angle \( \beta \) (degrees, between \( a \) and \( c \)):

Angle \( \gamma \) (degrees, between \( a \) and \( b \)):

Base Side \( a \) (cm):

Base Side \( b \) (cm):

Height \( h \) (cm, perpendicular to base):

Volume (cm³): Volume (in³): Volume (ft³): Volume (m³):

Surface Area (cm²): Surface Area (in²): Surface Area (ft²): Surface Area (m²):

1. What is a Parallelepiped Volume Calculator?

Definition: This calculator computes the volume and surface area of a parallelepiped using three different methods: vectors, sides and angles, or sides and height.

Purpose: Useful for geometric calculations in mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator offers three methods to compute the volume:

Method 1: Using Three Vectors

\[ V = |(\vec{a} \times \vec{b}) \cdot \vec{c}| \]

Compute the cross product \( \vec{a} \times \vec{b} \), then the dot product with \( \vec{c} \), and take the absolute value.

Method 2: Using Sides and Angles

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

Where \( a, b, c \) are adjacent sides, \( \alpha \) is the angle between \( b \) and \( c \), \( \beta \) is between \( a \) and \( c \), and \( \gamma \) is between \( a \) and \( b \).

Method 3: Using Sides and Height

\[ V = a \cdot b \cdot h \]

Where \( a \) and \( b \) form the base, and \( h \) is the height perpendicular to the base.

Surface Area (All Methods)

\[ A = 2 \cdot (|\vec{a} \times \vec{b}| + |\vec{b} \times \vec{c}| + |\vec{a} \times \vec{c}|) \]

Unit Conversions:

Input (Length): mm (×0.1), cm (×1), m (×100), in (×2.54), ft (×30.48)
Volume: cm³ (direct), in³ (×0.0610237), ft³ (×0.0000353147), m³ (×0.000001)
Surface Area: cm² (direct), in² (×0.155), ft² (×0.00107639), m² (×0.0001)

Explanation: Inputs are converted to cm, volume is calculated in cm³ and converted to in³, ft³, m³, and surface area is calculated in cm² and converted to in², ft², m².

3. Importance of Parallelepiped Volume Calculation

Details: Calculating the volume and surface area of a parallelepiped is essential in geometry, physics, and engineering applications.

4. Using the Calculator

Tips: Select a method, then enter the required inputs (vectors, sides and angles, or sides and height) in mm, cm, m, in, or ft (all >0, angles 0-180°). Results show the volume (cm³, in³, ft³, m³) and surface area (cm², in², ft², m²).

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