Parallelepiped

Parallelepiped is a 3-D shape whose faces are all parallelograms. It is obtained from a Greek word which means ‘an object having parallel plane’. Basically, it is formed by six parallelogram sides to result in a three-dimensional figure or a Prism, which has a parallelogram base. We can define it as a polyhedron, where three pairs of parallel faces are joined together to form a three-dimensional shape, having six faces. The cube, cuboid and rhomboid are its three special cases. The rectangular parallelepiped has all the faces in a rectangular shape.

Properties of Parallelepiped

• It is a three-dimensional solid figure.
• Any three faces can be viewed at the same time.
• It has three sets of four parallel edges and the edges within each set have equal measurement of length.
• The diagonal of each face is called face diagonal.
• On observing from outside, each face seems the mirror image of the opposite face.
• It signifies a Prism of parallelogram base.
• It is a polyhedron of six faces.
• The three pairs of parallel faces form a hexahedron.

Volume and Surface Area Formula

For a given parallelepiped, let S is the area of the bottom face and H is the height, then the volume formula is given by;

Since the base of parallelepiped is in the shape of a parallelogram, therefore we can use the formula for the area of the parallelogram to find the base area.

• Area of Parallelogram = Length × Height

Lateral Surface Area (LSA): Product of perimeter of the base and the height of the 6 parallelograms faced prism.

Total Surface Area (TSA): Addition of Lateral surface area and twice the base area

Rectangular Parallelepiped

When all the six faces of parallelepiped are in a rectangular shape, then it is considered a rectangular parallelepiped. It is a three-dimensional box-shaped structure. The length of all the parallel edges here are equal. The base of the prism here is rectangular in shape. A common example you can see in real life is the shoe box, which has a rectangular shape.

Here, the surface area is equal to the area of rectangle = Length × Width

Therefore, the volume becomes;

V = Length × Width × Height

So, if we know these three dimensions of the rectangular box, we can find its volume. Suppose, length = a, width = b and height = c, we can write the formula of volume, surface area and length of the diagonal of the rectangular box as;

Example

The base face of a parallelepiped has opposite sides measuring 5 inches and 10 inches. The height of the parallelepiped is 4 inches. Find the cost of painting its walls from outside at a cost of INR 1.5 per square inch.

Solution: We need to find the lateral surface area first, therefore;

LSA = Perimeter of base × height

LSA = 2 (5 + 10) × 6

LSA = 180 sq.inch

Cost of painting = Lateral surface area × cost per square inch

Cost of painting the walls = 180 × 1.5 = Rs.270/-

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