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Parallelepiped is a three-dimensional shape that has all parallelogram faces. The word parallelepiped is from a Greek word that means ‘an object having a parallel plane’. Parallelepiped is formed by area of parallelogram resulting in a three-dimensional figure or a Prism, which has a parallelogram base. The rectangular parallelepiped is a special type of parallelepiped that has all the faces in a rectangular shape. All sides are at a right angle with the adjoining side. The length of all the parallel edges or sides in a rectangular parallelepiped are equal.
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Key Takeaways: Rectangular parallelepiped; parallelepiped; rectangle; Menstruation; Volume; Surface area; Diagonal
Rectangular parallelepiped
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When all the six faces of parallelepiped are in a rectangular shape, the structure is considered as a rectangular parallelepiped. It is a three-dimensional box-shaped figure. The length of all the parallel edges or sides in a rectangular parallelepiped are equal. A common example in our daily life is a shoe box, which has a rectangular parallelepiped shape.
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Properties of rectangular parallelepiped
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There are certain properties of a rectangular parallelepiped by which we can distinguish it from other 3-D shapes. These properties are listed below:
- Rectangular parallelepiped is a 3-D solid shape.
- Rectangular parallelepiped has six faces, twelve edges, and eight vertices.
- All faces are perpendicular to each other.
- A rectangular parallelepiped has 2 diagonals on each face, called the face diagonals. It has a total of 12 face diagonals.
- The diagonals connecting the vertices not lying on the same face are called the body or space diagonal of a rectangular parallelepiped.
- Each face of a rectangular parallelepiped is a mirror image of the opposite face.
Rectangular parallelepiped formulas
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Following are the Rectangular Parallelepiped formulas:
Surface Area of Rectangular parallelepiped
The surface area of a rectangular parallelepiped is defined as the total area covered by all the surfaces of a parallelepiped. The surface area of a rectangular parallelepiped is expressed in square units, like in2, cm2, m2, etc. The surface area of rectangular parallelepiped can be of two types: Lateral Surface Area and Total Surface Area
Lateral Surface Area of Rectangular parallelepiped
The lateral surface area of a rectangular parallelepiped is defined as the area of the lateral or side faces of a rectangular parallelepiped. To calculate the LSA of a rectangular parallelepiped, we need to find the sum of the area covered by the 4 side faces.
| LSA of Parallelepiped = 2ab+ 2bc |
Total Surface Area of Rectangular parallelepiped
The total surface area of a rectangular parallelepiped is the summation of the area of all the faces of the rectangular parallelepiped. To calculate the TSA of a rectangular parallelepiped, we need to find the sum of the area covered by the 6 faces.
| TSA of Parallelepiped = 2ab+2bc+2ca |
Volume of Rectangular parallelepiped
The volume or mass of a rectangular parallelepiped is defined as the space occupied by the shape in a three-dimensional plane. The volume of a rectangular parallelepiped is expressed in cubic units, like in3, cm3, m3, etc. Volume of rectangular parallelepiped can be calculated using the base area and the height. The formula to calculate the volume of a rectangular parallelepiped is given as:
| Volume of rectangular parallelepiped = a× b× c |
Diagonal of Rectangular parallelepiped
The length of diagonal of a rectangular parallelepiped is the distance between the opposite vertices or corners.
Applying Pythagorean Theorem, the length of a diagonal of the lower face can be found out-
Since in rectangular parallelepiped all sides, as well as the faces, are perpendicular to each other, then x is perpendicular to c.
again, by Pythagorean Theorem, the length of a diagonal of a rectangular parallelepiped is:
| D2=X2+ C2 D2= a2+b2+c2 = Diagonal of rectangular parallelepiped |
here,
a= length
b= breadth
c= height
d= diagonal
Points to Remember
Following are some important points:
- Rectangular parallelepiped is a kind of parallelepiped with rectangular faces.
- All sides as well as the faces of a rectangular parallelepiped are at a right angle to each other.
- The diagonals connecting the vertices not lying on the same face are called the body or space diagonal of a rectangular parallelepiped.
- There are eight diagonals in rectangular parallelepiped.
Sample Questions
Ques: If the base face of a rectangular parallelepiped has opposite sides measuring 6 inches and 10 inches and its height is 7 inches, find the lateral surface area of the rectangular parallelepiped. (3 Marks)
Ans: Using the lateral area of rectangular parallelepiped formula,
LSA of Parallelepiped = 2ab+ 2bc
= 2*6*7+ 2*10*7
=84+ 140
= 224 sq inches
Lateral area of given parallelepiped = 224 in2.
Ques: A gift is packed in a rectangular parallelepiped box of dimensions 10 in, 7 in, and 8 in and it needs to be wrapped with gift paper. Find the area of gift paper required to wrap the gift box? (3 Marks)
Ans: The dimensions of the given gift box are,
length, l = 10 in
width, w = 7 in
height, h = 8 in
Since box is in a rectangular parallelepiped shape,
TSA = 2 (lw + wh + hl) = 2 (10 × 7 + 7 × 8 + 8 × 10)
= 2 (70 + 56 + 80)
= 412 in2.
The amount of area of the gift paper required = 412 in2.
Ques: The base face of a rectangular parallelepiped has opposite sides measuring 5 inches and 10 inches. The height of the rectangular parallelepiped is 4 inches. What is the cost of painting its walls from outside at INR 2 per square inch? (3 Marks)
Ans: We need to find the lateral surface area of rectangular parallelepiped first, therefore;
LSA = 2ab+ 2bc
LSA = 2× 5× 6 +2 × 10× 6
LSA = 180 sq. inch
Cost of painting = Lateral surface area × cost per square inch
Cost of painting the walls = 180 × 2 = Rs.360/-
Ques: Counting 38 cu. ft. of bitumen to a ton, how many tons will a bitumen bin 19 ft. long, 6 ft. wide, and 9 ft. deep contain, when level full? (4 Marks)
Ans: The volume of a rectangular parallelepiped:
V=L×W×H
V=(19ft) (6R.) (9ft.)
V=1026R3
Since, density is given by ρ=W/V [ ρ - density, W - weight, and V- volume]
Thus, the weight of bitumen in a bin is W = ρV
W = 1026R3(1 ton/38 r3)
W = 27 tons
Ques: External dimensions of a wooden rectangular parallelepiped are 30 cm × 25 cm × 20 cm. If the thickness of the wood is 2 cm all around, find the volume of the wood contained in the rectangular parallelepiped formed. (4 Marks)
Ans: External length of the rectangular parallelepiped = 30 cm
External breadth of the rectangular parallelepiped = 25 cm
External height = 25 cm
External volume = (30 × 25 × 20) cm³
= 15000 cm³
Internal volume = (26 × 21 × 16) cm³
= 8736 cm³
Hence, Volume of the wood contained = External Volume - Internal Volume
= (15000 – 8736) cm³ = 6264 cm³
These are the above step-by-step detailed explanation in calculating worked-out problems on volume of a rectangular parallelepiped
Ques: Find the lateral and total surface area of a rectangular parallelepiped of length 80 cm, breadth 40 cm and height 20 cm. (4 Marks)
Ques: The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. What is the cost of whitewashing the walls and the ceiling of the room at the rate of Rupees 50 m²? (4 Marks)
Ans: Since the room is of rectangular parallelepiped shape
Length of a room (l) = 5m
Breadth (b) = 4 m
and height (h) = 3 m
∴ Area of 4 walls = 2(l + b) x h
= 2(5 + 4) x 3 = 6 x 9 = 54 m²
Area of ceiling = l x b = 5 x 4 = 20 sq metre
Total area = 54 + 20 = 74 sq metre
Rate of whitewashing = 7.50 per m²
Total cost = Rupees 74 x 7.50 = Rupees 555
Ques: Find the cost of digging a rectangular parallelepiped pit 8 m long, 6 m broad and 3 m deep at the rate of â?¹30 per m3. (3 Marks)
Ans: Length of pit (l) = 8m
Width (b) = 6 m
and depth (h) = 3 m
∴ Volume of earth dig out = lbh = 8 x 6 x 3 = 144 m3
Given, Cost of digging the pit at the rate of rupees 30 per m3
= 144 x 30 = â?¹4320
Ques: Dimensions of a cinema hall are 100 m, 50 m and 18 m. How many people can sit in the hall, if each person needs 150 m3 of air? (4 Marks)
Ans: Since the hall is of rectangular parallelepiped shape
Length of hall (l) = 100 m
Breadth (b) = 50 m
and height (h) = 18 m
Volume of air in it = lbh = 100 x 50 x 18 = 90000 cubic metres
The air required for one person = 150 m3
So, Number of persons in the hall = 90000/ 150 = 600
Ques: Two cubes of the side 6 cm are joined to form a cuboid. Find the total surface area of the cuboid. (3 Marks)
Ans: When two cubes are joined end to end, then
Length of the cuboid = 6 + 6 = 12 cm
Breadth of the cuboid = 6 cm
Height of the cuboid = 6 cm
Total surface area of the cuboid = 2 (lb + bh + hi)= 2(12 x 6 + 6×6 + 6×12)
= 2(72 + 36 + 72) = 2(180) = 360 cm2
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