A parallelepiped is a three-dimensional solid shape that looks like a box with six sides, where each side is a parallelogram. It is a type of prism with a parallelogram as its base. Three-dimensional shapes, or 3D shapes, have length, width, and height, which are used to measure them. In this topic, we will learn about the parallelepiped, its key properties, and the formulas to find its total surface area, lateral surface area, and volume. We will also look at the equation for a rectangular parallelepiped and solve examples to understand how to apply these formulas in real problems.

What is a Parallelepiped?

A parallelepiped is a three-dimensional figure formed by six parallelograms and it can be defined as a polyhedron with six faces also called a hexahedron with each face a parallelogram. A parallelepiped is also called a rhomboid. By analogy with 2D Shapes, it correlates to a parallelogram just as a cube relates to a square. The rectangular cuboid having six rectangular faces, a cube having six square faces, and the rhombohedron with six rhombus faces are all specific cases of the parallelepiped.

Volume of Parallelepiped

The volume of the parallelepiped is the amount of space a parallelepiped contains. To understand this consider a parallelepiped filled with water. The volume of water that fits in the parallelepiped is the volume of the parallelepiped.

The volume of the parallelepiped is the area of the base times the height. From the geometric definition of the cross product, we know that its magnitude, ∥a×b∥, is the area of the parallelogram base, and that the direction of the vector a×b is perpendicular to the base. To describe a parallelepiped, we need its three adjacent sides and the measure of the angles between them, or the three adjacent vectors.

Volume of Parallelepiped Formula: Formula for the volume of a parallelepiped formed by vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) with the same initial point is given by their scalar triple product:

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

V – Volume of the parallelepiped formed by the three vectors; \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\)

The first step of calculating the volume of the parallelepiped involves finding the cross product or vector product of \((\vec{a}\times\vec{b})\). The product vector is perpendicular to both \(\vec{a}\) and \(\vec{b}\). Note that the magnitude of the resultant vector, \(\lvert\vec{a}\times\vec{b}\rvert\), is equal to the area of a parallelogram described by these two vectors.

The subsequent dot product between \((\vec{a}\times\vec{b})\) and \(\vec{c}\) denotes the projection of \(\vec{a}\times\vec{b}\) on \(\vec{c}\). In other words, it sweeps the base parallelogram along \(\vec{c}\), analogous to multiplying the base area with height.

We can further simplify the formula and reduce it to one determinant:

\(\begin{matrix}
(\vec{a}\times\vec{b})\cdot\vec{c}
&=
\left|
\begin{array}{cc}
a_2 & a_3\\
b_2 & b_3
\end{array}
\right|
c_1
–
\left|
\begin{array}{cc}
a_1 & a_3\\
b_1 & b_3
\end{array}
\right|
c_2
+
\left|
\begin{array}{cc}
a_1 & a_2\\
b_1 & b_2
\end{array}
\right|
c_3
\\
&=
\left|
\begin{array}{ccc}
c_1 & c_2 & c_3\\
a_1 & a_2 & a_3\\
b_1 & b_2 & b_3
\end{array}
\right|.
\end{matrix}\)

This gives us the volume of the parallelepiped when the vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are known.

Volume of Parallelepiped using Vertices

If we know the vertices of a parallelepiped we can find out its vectors by using the simple distance formula. If \(A < x_1, y_1, z_ 1 >, B < x_2, y_2, z_ 2 >\), \(C < x_3, y_3, z_ 3 >\) and \(D < x_4, y_4, z_ 4 >\) then we can find the vectors AB =a, AC=b and AD=c sharing the same initial point and use them as our basis vectors generating the parallelepiped. AB will be given by \( a = < x_2 – x_1, y_2 – y_1, z_2 – z_1 >\), Similarly we can find AC and AD.

Once we have vectors a, b, and c, follow these simple steps: Find the cross-product between the vectors \(\vec{a}\) and \(\vec{b}\) to get \(\vec{a}\times\vec{b}\).

Calculate the dot-product between the vectors \(\vec{a}\times\vec{b}\) and \(\vec{c}\) to get the scalar value \((\vec{a}\times\vec{b})\cdot\vec{c}\).

Determine the volume of the parallelepiped as the absolute value of this scalar, given by \(∣(\vec{a}\times\vec{b})\cdot\vec{c}∣\).

Volume of Parallelepiped Using Sides and Angles

Sometimes you don’t know the vertices of the parallelepiped but the angles and side lengths are known:

To calculate the volume of a parallelepiped from its sides and angles we use the formula:

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

Where,

\(\alpha\) – Angle between the sides b and c;

\(\beta\) – Angle between the sides a and c; and

\(\gamma\) – Angle between the sides a and b.

Area of Parallelepiped

Area of Parallelepiped is the sum of the area of all its surfaces. This is called the total surface area of the parallelepiped. When it comes to area 3D objects there are two important surface areas:

• Total Surface Area of Parallelepiped
• Lateral Surface Area of Parallelepiped

Learn about the area of quadrilateral.

Lateral Surface Area of a Parallelepiped

The lateral surface area of a parallelepiped is defined as the sum of the surface area of its faces excluding the base and the top. The Lateral Surface Area (LSA) of a Parallelepiped is equal to the product of the perimeter of the base and the height of the Parallelepiped.

Lateral surface area of a parallelepiped Formula: \(LSA= 2(a+b){\times}c\)

Total Surface Area of a Parallelepiped

The total surface area of a parallelepiped is defined as the total region covered by all the faces of the shape.

Total surface area of a parallelepiped Formula: \(TSA=2(ab+bc+ac)\)

In vector form, we can write it as,

\(A = 2{\times}(∣\vec{a}{\times}\vec{b}∣ + ∣\vec{b}{\times}\vec{c}∣ + ∣\vec{a}{\times}\vec{c}∣)\)

Where,

\(∣\vec{a}{\times}\vec{b}|: Magnitude \; of \; the \; cross-product \; between \; \vec{a}\) and \(\vec{a}\)

\(∣\vec{b}{\times}\vec{c}|: Magnitude \; of \; the \; cross-product \; between \; \vec{b}\) and \(\vec{c}\)

\(∣\vec{c}{\times}\vec{a}|: Magnitude \; of \; the \; cross-product \; between \; \vec{c}\) and \(\vec{b}\)

Area of a Parallelepiped using Sides and Angles

There is an alternative way to calculate the area of a parallelepiped using its side lengths \(a\), \(b\) and angles. It is given by the formula

\(A = 2\times(a{\cdot}b{\cdot}sin(\gamma) + b{\cdot}c{\cdot}sin(\alpha) + a{\cdot}c{\cdot}sin(\beta))\), where:

\(\alpha\) – Angle between b and c;

\(\beta\) – Angle between a and c; and

\(\gamma\) – Angle between a and b.

Rectangular Parallelepiped

A rectangular parallelepiped is a polyhedron with six faces, which is also known as a hexahedron. It is a three-dimensional parallelepiped structure whose all six faces are in rectangular shape and the length of the parallel edges are equal. It is a special case of the parallelepiped. A rectangular parallelepiped is analogous to a 2D rectangle.

Learn about rectangles.

Rectangular Parallelepiped Formula

The formula of a rectangular parallelepiped includes the formula for its surface area, volume and diagonals.

Surface Area of Rectangular Parallelepiped

There are six rectangles in the Rectangular Parallelepiped figure. We must calculate the area of six rectangles in order to determine their surface area (faces). The surface area formula is given by:

Surface Area \(= 2(l{\times}h) + 2(l{\times}w) + 2(h{\times}w)\)

\(S = 2[(l{\times}h) + (l{\times}w) + (h{\times}w)]\)

where

\(l\), \(w\), \(h\) are the length, width, height of the rectangular parallelepiped respectively.

Lateral Surface Area of Rectangular Parallelepiped

The product of base perimeter and height is known as lateral surface area. Each face of a rectangular parallelepiped figure is a rectangle, making the base’s perimeter equal to the rectangle’s perimeter. The following is the formula for LSA (Lateral Surface Area):

LSA = Perimeter of base × Height

As the perimeter of the base is equal to 2(length + breadth)

= 2(length + breadth) × Height

\(LSA = 2lh + 2wh\)

where

\(l\), \(w\), \(h\) are length, width and height of the rectangular parallelepiped respectively.

Volume of Rectangular Parallelepiped

The area of the base multiplied by the height yields the volume of a rectangular parallelepiped. The base of a rectangular parallelepiped is a rectangle, just like each of its faces, and its area is equal to the product of its length and width. The volume formula is as follows:

Volume = area of base × height

\(V = l{\times}w{\times}h\)

Diagonal Length of Rectangular Parallelepiped

Let \(d\) be the length of the body diagonal of a rectangular parallelepiped having length, width and height as \(l\), \(w\) and \(h\), then this length is given by:

\(d=\sqrt{l^2+w^2+h^2}\)

Characteristics of a Parallelepiped 

1. Six Parallelogram Faces: A parallelepiped has six faces, and each face is a parallelogram.
2. Parallel Faces: Opposite faces are parallel to each other and equal in size.
3. Edges and Vertices: It has 12 edges and 8 vertices. The edges are grouped in three sets of four, with edges in the same set equal in length.
4. Analogy to 2D Shape: A parallelepiped in 3D is similar to a parallelogram in 2D, just as a cube is related to a square.
5. Face Diagonals: Each face has two diagonals, which help in calculating lengths and angles.
6. Mirror Images: Opposite faces appear as mirror images of each other.
7. Prism Shape: It is a type of prism with a parallelogram as its base.
8. Hexahedron: It has six flat faces, making it a type of hexahedron.
9. Polyhedron: Being made of flat polygonal faces, it is a polyhedron.
10. Angles Between Faces: Depending on the type of parallelepiped, the angles between adjacent faces can be right angles or oblique.
11. Volume and Space: It occupies space in three dimensions and its volume can be calculated using the base area and height.

Properties of Parallelepiped 

1. A parallelepiped is a three-dimensional solid shape.
2. You can see any three of its faces at the same time.
3. It has three sets of edges, with four edges in each set. All edges in the same set are equal in length.
4. The diagonal of each face is called a face diagonal.
5. Each face looks like a mirror image of the face opposite to it.
6. It is a type of prism with a parallelogram as its base.
7. A parallelepiped has six flat faces, making it a polyhedron.
8. The three pairs of parallel faces together form a hexahedron.

Applications of Parallelepiped 

1. Mathematics and Geometry – Parallelepipeds help calculate volume, surface area, and diagonals in 3D geometry.
2. Physics and Engineering – Used to study forces, mass, and weight distribution in objects like beams and blocks.
3. Architecture and Construction – Many buildings, rooms, and storage containers are designed as rectangular parallelepipeds for strength and space efficiency.
4. Computer Graphics and 3D Modelling – Parallelepipeds form the base for 3D objects in virtual simulations and help calculate intersections and shading.
5. Daily Life Objects – Many common items like boxes, books, bricks, and refrigerators are shaped as parallelepipeds for easy stacking and storage.

Solved Examples on Parallelepiped

Some important solved examples on Parallelepiped are given below:

Example 1: Find the total surface area of the given parallelepiped if length is 13 cm, breadth is 12 cm and height is 10 cm.

Solution: We know that,

Total Surface Area (TSA) = 2(ab + bc + ac), here a = 13 cm, b = 12 cm, c = 10 cm

= 2(13 × 12 + 12 × 10 + 10 × 13 )

= 812 sq.cm

Example 2: Consider a parallelepiped ABCDEFGH with edge lengths a = 7, b = 5, and c = 3. If ∠DAE=45°, ∠BAD=63°, and ∠BAE=50°, then what is the surface area of the parallelepiped?

Solution: Here,

\(\alpha=∠BAD=63°\) – Angle between b and c;

\(\beta=∠DAE=45°\) – Angle between a and c; and

\(\gamma=∠BAE=50°\) – Angle between a and b.

The surface area will be given by, would be given by

\(A = 2\times(a{\cdot}b{\cdot}sin(\gamma) + b{\cdot}c{\cdot}sin(\alpha) + a{\cdot}c{\cdot}sin(\beta))\)

\(A = 2\times(7{\cdot}5{\cdot}sin(50) + 5{\cdot}3{\cdot}sin(63) + 7{\cdot}3{\cdot}sin(45))\)

\(A = 2\times(35{\cdot}sin(50) + 15{\cdot}sin(63) + 21{\cdot}sin(45))\)

\(A = 110.05\) sq. units

Example 3: Find the volume of a parallelepiped with vectors a = (1, 1, 2), b = (2, 1, 3) and c = (3, 1, 2).

Solution: Volume of a parallelepiped is given by the determinant of the matrix:

\( V=
{\begin{bmatrix}
c_1 & c_2 & c_3\\
a_1 & a_2 & a_3\\
b_1 & b_2 & b_3
\end{bmatrix}}
\)

Substituting the values we get

\( V=
{\begin{bmatrix}
3 & 1 & 2\\
1 & 1 & 2\\
2 & 1 & 3
\end{bmatrix}}
\)

Finding the determinant we get,

\(V = |3(1\times3 – 2\times1) – 1(1\times3 – 2\times2) + 2(1\times1 – 1\times2)|\)

\(V = |3\times1 – 1\times(-1) + 2\times(-1)|\)

\(V = 2 \) cube units.

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