Rectangular Parallelepiped Formula

Rectangular Parallelepiped Formula

A rectangular parallelepiped, also known as a rectangular prism or cuboid, is a three-dimensional geometric figure bounded by six rectangular faces. Each pair of opposite faces is parallel and congruent, meaning they are identical in shape and size. The edges of the rectangular parallelepiped are perpendicular to each other, and the faces meet at right angles. This solid has twelve edges, eight vertices, and six faces. It can be characterized by three dimensions: length, width, and height.

The formulas for a rectangular parallelepiped are used to calculate its volume, surface area, and diagonal. Learn more about the rectangle parallelepiped, its definition, and its formula in this article.

What is Parallelepiped?

A parallelepiped is a three-dimensional geometric figure with six faces, each of which is a parallelogram. It is a type of polyhedron and is defined by the parallelism of its opposite faces. Here are the key characteristics and properties of a parallelepiped:

1. Faces: A parallelepiped has six faces, with each face being a parallelogram. Opposite faces are parallel and congruent to each other.
2. Edges: It has twelve edges, grouped into three sets of four parallel edges. Each set of edges is parallel to one another and of equal length.
3. Vertices: The parallelepiped has eight vertices, where three edges meet at right angles, though in the general case, the angles between edges can vary.

Rectangular Parallelepiped

A rectangular parallelepiped, also known as a rectangular prism or a cuboid, is a three-dimensional geometric figure with six rectangular faces. It is a special type of parallelepiped where all angles between adjacent faces are right angles (90 degrees). This makes it one of the most common and easily recognizable shapes in both mathematics and everyday life.

Properties of a Rectangular Parallelepiped

1. Faces:
   • It has six rectangular faces.
   • Each face is a rectangle, and opposite faces are congruent (identical in shape and size).
2. Edges:
   • It has twelve edges.
   • These edges can be grouped into three sets of four parallel edges.
   • Each set of edges corresponds to one of the three dimensions: length, width, and height.
3. Vertices:
   • It has eight vertices.
   • At each vertex, three edges meet, and the angles between these edges are right angles.
4. Right Angles:
   • All interior angles are right angles (90 degrees).
   • This right-angle property is what distinguishes a rectangular parallelepiped from a more general parallelepiped.
5. Dimensions:
   • It is defined by three dimensions: length (l), width (w), and height (h).
   • These dimensions correspond to the edges meeting at a vertex

Formula of Rectangular Parallelepiped

The formula of rectangular parallelepiped include mathematical expressions to calculate volume, surface area and diagonal of rectangular parallrlepiped

Rectangular Parallelepiped Volume Formula

The volume of a rectangular parallelepiped (also known as a rectangular prism or cuboid) can be calculated using a straightforward formula. A rectangular parallelepiped is defined by its three dimensions: length (l), width (w), and height (h). The volume \( V \) is given by the product of these three dimensions.

The formula for the volume \( V \) of a rectangular parallelepiped is:

\[ V = l \times w \times h \]

where:
\( l \) is the length of the parallelepiped,
\( w \) is the width of the parallelepiped,
\( h \) is the height of the parallelepiped.

Rectangular Parallelepiped Surface Area Formula

The surface area of rectangular parallelepiped are of two types

• Lateral Surface Area (LSA): This is the sum of the areas of the four vertical faces (assuming the parallelepiped is standing on one of its faces).
• Total Surface Area (TSA): This includes the areas of all six faces of the parallelepiped.

Lateral Surface Area (LSA)

The LSA is the sum of the areas of the four vertical faces. If we assume$ा$ l to be length, and$b$ b to be breadth and h is the height, the LSA is given by:

LSA=2(lh+bh)

Total Surface Area (TSA)

The TSA includes the areas of all six faces. The formula is:

TSA=2(lb+bh+hl)

Rectangular Parallelepiped Diagonal Formula

The diagonal of a rectangular parallelepiped (rectangular prism) can be calculated using the Pythagorean theorem in three dimensions. Given the edge lengths \( a \), \( b \), and \( c \), the formula for the length of the space diagonal \( d \) is:

\[ d = \sqrt{a^2 + b^2 + c^2} \]

Solved Examples on Rectangular Parallelepiped Formula

Example 1: Find the volume of a rectangular parallelepiped with length 5 units, width 3 units and height 4 units.

Solution:

Let’s consider a rectangular parallelepiped with the following dimensions:
Length (\( l \)) = 5 units
Width (\( w \)) = 3 units
Height (\( h \)) = 4 units

To find the volume, we substitute these values into the formula:

\[ V = l \times w \times h \]
\[ V = 5 \times 3 \times 4 \]
\[ V = 60 \]

So, the volume of this rectangular parallelepiped is 60 cubic units.

Example 2: Find LSA and TSA of a rectangular parallelepiped if given dimensions are

Length ( a) = 3 units
Width ( b) = 4 units
Height ( c) = 5 units

Solution:

1. Lateral Surface Area (LSA):

\[ \text{LSA} = 2(ac + bc) \]
\[ \text{LSA} = 2(3 \cdot 5 + 4 \cdot 5) \]
\[ \text{LSA} = 2(15 + 20) \]
\[ \text{LSA} = 2 \cdot 35 \]
\[ \text{LSA} = 70 \text{ square units} \]

2. Total Surface Area (TSA):

\[ \text{TSA} = 2(ab + bc + ca) \]
\[ \text{TSA} = 2(3 \cdot 4 + 4 \cdot 5 + 5 \cdot 3) \]
\[ \text{TSA} = 2(12 + 20 + 15) \]
\[ \text{TSA} = 2 \cdot 47 \]
\[ \text{TSA} = 94 \text{ square units} \]

Example 3: Find the diagonal of a rectangular parallelepiped if the given dimensions are

Length (a) = 3 units
Width (b) = 4 units
Height (c) = 5 units

Solution:

The formula for the length of the diagonal \( d \) of a rectangular parallelepiped is:

\[ d = \sqrt{a^2 + b^2 + c^2} \]

Substitute the given values into the formula:

\[ d = \sqrt{3^2 + 4^2 + 5^2} \]
\[ d = \sqrt{9 + 16 + 25} \]
\[ d = \sqrt{50} \]
\[ d = \sqrt{25 \cdot 2} \]
\[ d = 5\sqrt{2} \]

So, the length of the diagonal \( d \) is \( 5\sqrt{2} \) units.

These examples illustrate how to apply the surface area formulas for rectangular parallelepipeds using given dimensions.