Parallelpiped

High School

Definition

In geometry, it is a polyhedron with 6 faces made up of parallelograms. Much like how a parallelogram looks like a slanted square, a parallelpiped is like a slanted cube. Its volume can be calculated by multiiplying the area of the base with the height. The formula is a · (b × c) where each letter desinates each edge on a different plane.

Worked examples

\(V = \)base area\( \times h\)

If the base parallelogram has area 12 and height is 5, then volume is \(12 \times 5 = 60\) cubic units.

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

The scalar triple product of three edge vectors gives the signed volume of the parallelpiped.

Common mistakes

\(V = l \times w \times h\) using slant edges → \(V = \)base area\( \times h\) where \(h\) is perpendicular height You must use the perpendicular height, not the slant edge length.
Treating all six faces as rectangles → All six faces are parallelograms (rectangles only if edges are perpendicular) A parallelpiped is slanted; only special cases (rectangular prisms) have all rectangular faces.
\(\vec{a} \times (\vec{b} \times \vec{c})\) for volume → \(\vec{a} \cdot (\vec{b} \times \vec{c})\) Volume uses the dot product of one vector with the cross product of the other two, not a double cross product.

Where you'll use it next

Parallelepipeds appear in multivariable calculus when computing volumes via the scalar triple product, in linear algebra for understanding determinants and transformations, and in physics for modeling crystal lattices and stress tensors.

Found in 1 StudyPug lesson

Surface Area and Volume of Prisms: Essential Geometry Concepts

Geometry

Dive into the world of prisms! Learn to calculate surface area and volume, master key formulas, and apply your skills to real-world problems. Boost your geometry prowess with our comprehensive guide.

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