In the day to day life, people encounter different objects of varied shapes and sizes, starting from mobile phones, laptops, LPG cylinders, etc. to trucks, buildings, dams. Every object has a definite shape owing to its usage or the utility it would create for us. It is obvious that the dimensions of these objects are predetermined as well, owing to the needs of the people using them. The study of all these shapes and the measurement of such dimensions is necessary for the objects to be able to be put to use readily.
For example, a water tank to be installed on a rooftop needs to have significant capacity or volume as per the needs of the user(s). Depending on the capacity, the dimensions, i.e., length, breadth, and height can be customized. This is where the concept of mensuration comes into the picture.
Mensuration
The study of various dimensions or proportions pertaining to different geometrical shapes is termed mensuration in mathematics. Mensuration offers a wide variety of formulas to calculate various quantities pertaining to these shapes, thus having its significance not only in mathematical theory but also in our daily lives. There are certain shapes that have only two dimensions and those which have three dimensions. They are segregated because the measurements pertaining to them vary on the basis of the number of proportions. They are 2-D shapes, like the circle, square, triangle, etc., and 3-D shapes like cube, cuboid, cylinder, cone, etc.
Cuboid
A cuboid can be defined as a three-dimensional shape that is composed of six rectangles as its faces, having all its vertices at 90° angles. The number of faces, edges, and faces in a cuboid is 6, 12, and 8 respectively. None of the edges are of the same length in the case of a cuboid. At least four of the faces of a shape have to be identical for it to be called a cuboid. Such a cuboid whose all angles are right angles, and opposite faces are equal is called a rectangular cuboid. Mobile phones, microwaves, water tanks, books, boxes, elevators, refrigerators, etc. are some daily life examples of the shape called cuboid.
The following figure depicts a cuboid with two faces at the top and bottom, front and back, and two on either side. It is to be noted how the opposite faces are equal to each- other, but no two adjacent faces are equal.
Properties of a Cuboid
- A cuboid has 6 faces, all of which are rectangular in shape.
- The angles forming at all the vertices of a cuboid measure 90 degrees each.
- A cuboid has three dimensions, namely length, width, and height.
- On each face of a cuboid, two diagonals can be drawn which would then intersect each- other.
- Opposite edges of a cuboid do not intersect each – other, i.e., are parallel.
Surface Area of a Cuboid
- Total Surface Area: Since a cuboid is constituted of six rectangular faces, the total surface area of a cuboid would be equal to the total area occupied by its six faces. Now,
- Total area of front and back face = 2(length × height) = 2lh
- Total area of the sideward faces = 2(height × breadth) = 2bh
- Total area of top and bottom faces = 2(length × breadth) = 2lb
TSA of a cuboid = 2 lh + 2bh + 2 lb = 2(lh + bh + lb)
- Lateral Surface Area: The area of a cuboid that is obtained by not taking into account its top and bottom faces is called its lateral or curved surface area. Thus,
- Total area of front and back face = 2(length × height) = 2 lh
- Total area of the side- ward faces = 2(height × breadth) = 2bh
LSA of a cuboid = 2 lh + 2bh = 2(lh + bh)
Volume of a Cuboid
The product of all three dimensions of a cuboid yields its volume. Volume represents the space occupied by the dimensions of a cuboid inside it.
Volume of a cuboid = l × b × h.
Find the height of a cuboid whose volume is 275 cm3 and the base area is 25 cm2.
Solution:
Let the length, breadth and height of the cuboid be depicted by l, b and h respectively.
Given: Volume = l × b × h = 275 cm3 ⇢ (1)
and, Base area = l × b = 25 cm2 ⇢ (2)
Substituting (2) into (1),
25h = 275
h = 275/ 25 cm
h = 11 cm
Thus, the height of the cuboid is 11 cm.
Similar Problems
Question 1: Find the height of a cuboid whose volume is 1000cm3 and base area is 100 cm2.
Solution:
Let the length, breadth and height of the cuboid be depicted by l, b and h respectively.
Given: Volume = l × b × h = 1000 cm3 ⇢ (1)
and, Base area = l × b = 100 cm2⇢ (2)
Substituting (2) into (1),
100h = 1000
h = 1000/ 100 cm
h = 10 cm
Thus, the height of the cuboid is 10 cm.
Question 2: Find the height of a cuboid whose volume is 180 cm3 and the base area is 90 cm2.
Solution:
Let the length, breadth and height of the cuboid be depicted by l, b and h respectively.
Given: Volume = l × b × h = 180 cm3 ⇢ (1)
and, Base area = l × b = 90 cm2 ⇢ (2)
Substituting (2) into (1),
90h = 180
h = 180/ 90 cm
h = 2 cm
Thus, the height of the cuboid is 2 cm.
Question 3: Find the height of a cuboid whose volume is 3600 cm3 and the base area is 600 cm2.
Solution:
Let the length, breadth and height of the cuboid be depicted by l, b and h respectively.
Given: Volume = l × b × h = 3600cm3 ⇢ (1)
and, Base area = l × b = 600 cm2 ⇢ (2)
Substituting (2) into (1),
60h = 3600
h = 3600/ 600 cm
h = 6 cm
Thus, the height of the cuboid is 6 cm.
Question 4. Find the height of a cuboid whose volume is 950 cm3 and the base area is 190 cm2.
Solution:
Let the length, breadth and height of the cuboid be depicted by l, b and h respectively.
Given: Volume = l × b × h = 950 cm3 ⇢ (1)
and, Base area = l × b = 190 cm2 ⇢ (2)
Substituting (2) into (1),
190h = 950
h = 950/ 190 cm
h = 5 cm
Thus, the height of the cuboid is 5 cm.
Question 5: Find the height of a cuboid whose volume is 343 cm3 and the base area is 49 cm2.
Solution:
Let the length, breadth and height of the cuboid be depicted by l, b and h respectively.
Given: Volume = l × b × h = 343 cm3 ⇢ (1)
and, Base area = l × b = 49 cm2 ⇢ (2)
Substituting (2) into (1),
49h = 343
h = 343/ 49 cm
h = 7 cm
Thus, the height of the cuboid is 7 cm.