Volume of a Parallelepiped Calculator
Calculate volume and surface area using vectors, vertices, or edge lengths
Calculate Parallelepiped Volume
Vector a = a₁i + a₂j + a₃k
Vector b = b₁i + b₂j + b₃k
Vector c = c₁i + c₂j + c₃k
Calculation Results
Volume
Surface Area
Vector Calculations
Example Calculations
Example 1: Three Vectors
Given: a = i + 2j + 3k, b = 5i - 4j + 7k, c = -5i + j + 12k
Cross Product: a × b = (26i + 8j - 14k)
Scalar Triple Product: (a × b) · c = 26(-5) + 8(1) + (-14)(12) = -290
Volume: |-290| = 290 cubic units
Example 2: Edge Lengths Method
Given: a = 5, b = 4, c = 7, α = 45°, β = 50°, γ = 63°
Formula: V = abc√(1 + 2cos(α)cos(β)cos(γ) - cos²(α) - cos²(β) - cos²(γ))
Calculation: V = 5 × 4 × 7 × √(1 + 2×0.707×0.643×0.454 - 0.5 - 0.413 - 0.206)
Volume: ≈ 75.83 cubic units
Example 3: Using Vertices
Given: P(0,0,0), Q(1,2,3), R(5,-4,7), S(-5,1,12)
Vectors: a = Q - P = (1,2,3), b = R - P = (5,-4,7), c = S - P = (-5,1,12)
Result: Same as Example 1, Volume = 290 cubic units
Volume Formulas
Scalar Triple Product
Determinant Method
Edge Lengths
Surface Area
Parallelepiped Properties
Shape: 6 parallelogram faces
3 pairs of parallel faces
Vectors: Three adjacent edges
Define entire solid
Volume = 0: Vectors coplanar
All vectors in same plane
Area = 0: Vectors collinear
All vectors on same line
Real-World Applications
Engineering
Structural analysis, material volumes
Crystallography
Unit cell volumes, crystal structures
Linear Algebra
Determinants, transformations
Computer Graphics
3D modeling, collision detection
Understanding Parallelepiped Volume
What is a Parallelepiped?
A parallelepiped is a three-dimensional figure formed by six parallelograms. It can be described by three vectors representing its adjacent edges. The volume represents the amount of space enclosed by this solid.
Scalar Triple Product
The volume is calculated using the scalar triple product: (a × b) · c. This combines the cross product (which gives the area of the parallelogram base) with the dot product (which projects the third vector onto the perpendicular direction).
V = |(a × b) · c|
Absolute value ensures positive volume
Calculation Methods
Vector Method: Uses three vectors directly
Most common for mathematical problems
Vertex Method: Calculates vectors from coordinates
Useful when given corner points
Edge Method: Uses lengths and angles
Practical for physical measurements
Special Cases
- • Volume = 0: Vectors are coplanar (flat shape)
- • Surface Area = 0: Vectors are collinear (line)
- • Negative scalar product: Vectors form left-handed system