Mensuration

Cube and Cuboid

When you look around the room in your house, you can see many objects that have the shape of a rectangular box. For example, furniture, books and TV are in the shape of a rectangular box. This shape is a cuboid. On the other hand, objects such as ice, dice and Rubik’s cube are examples of another 3D shape called a cube. As a matter of fact, a cube is a special type of cuboid in which all sides are squares and identical. Let us learn more about cubes and cuboids.

Suggested Videos

Play
Play
Play
previous arrow
next arrow
previous arrownext arrow
Slider

 

What is Cuboid?

Cuboid

The cuboid shape has a closed three-dimensional structure surrounded by rectangular faces, which are rectangle plane sections. It is one of the most prevalent shapes in our environment, with three dimensions: length, breadth, and height.

What is Area of Rectangle?

Faces, Vertices and Edges of Cuboid

  • Faces – A cuboid has 6 faces
  • Vertices – A cuboid has 8 vertices
  • Edges – A cuboid has 12 edges’

Properties of a Cuboid

  • Faces of a Cuboid 

A cuboid is made up of a total of 6 rectangles, and each rectangle is called a face. In the figure above, ABCD, DEFC, BCFG, ABGH, HEFG, ADEH are the 6 faces of cuboid. Two opposite faces in a cuboid are of equal lengths and areas. For example, ABCD and HGFE are two opposite faces. A cuboid’s faces are all rectangular in shape.

  • Edges of a Cuboid 

There are a total of 12 edges in a cuboid. They are AB, BC, CD, AD, DE, EF, FC, AH, HG, GB, HE, GF. In a cuboid, the opposite edges are of equal lengths. For example, AB=DC=HG=EF, AD=BC=HE=GF, AH=BG=DE=CF. Opposite edges in a cuboid are parallel to each other.

  • Vertices of a Cuboid 

A cuboid has 8 vertices. They are A, B, C, D, E, F, G, H. All of the angles created at a cuboid’s vertices are right angles.

  • Face Diagonal 

Face diagonals can be created by connecting the opposite vertices on a cuboid’s face. For example, AC is a face diagonal for the Face ABCD. There can be a total of 12 face diagonals in a cuboid.

  • Space Diagonal  

A space diagonal is a line segment that connects a cuboid’s opposite vertices. The space diagonals run through the cuboid’s interior. As a result, 4 space diagonals can be drawn within it. For example, HC is a space diagonal.

How to identify a cuboid?

In a cuboid, each face is a rectangle and the corners or the vertices are 90-degree angles. Also, the opposite faces are always equal. For example, a book is a cuboid. It has 6 surfaces of which each opposite pair is of the same dimensions.

Download Mensuration Cheat Sheet PDF Below

Total Surface Area of Cuboid

If l is the length, b is the breadth and h is the height of the cuboid, then the sum of areas of six rectangles of a cuboid gives the total surface area of the cuboid. The formula for it is given below.

Total surface area of a cuboid = 2 [( l × b ) + ( l× h ) + ( b× h )]

Browse more Topics under Mensuration

Lateral Surface Area of Cuboid

The sum of the area of 4 side faces i.e. leaving the bottom and the top face gives the lateral surface area of a cuboid. An example of the lateral surface area is the sum of the area of the four walls of a room. The formula to calculate the lateral surface area of a cuboid is

Area of four sides = 2 ( l × h ) +  2 ( b× h ) = 2 ( l + b ) × h = perimeter of base ×height

or simply,

Lateral surface area of a cuboid = 2(l+b)h

Volume of Cuboid

The volume of a cuboid can be found by multiplying the base area with the height. Therefore,

volume (V) = A  x h = (l x b) x h. In simple terms,

Volume of cuboid (v) = l × b × h

where l is the length, b is the base and h is the height of the cuboid.

Diagonal of a Cuboid

The length of the longest diagonal of a cuboid is given by

Length of diagonal of cuboid = √ (l² + b² + h²)

What is Area of Trapezium, Parallelogram, and Rhombus?

Examples of Cuboid

A few examples of cuboids in our daily lives are tall buildings, books, boxes, mobile phones, televisions, microwaves, photo frames, mattresses, bricks, etc.

TSA, LSA

Square units are used to represent the surface area of a cuboid.

Volume of a Cuboid

The volume of a cuboid is the amount of space occupied within a cuboid. The volume of a cuboid is determined by its length, width, and height. Cubic units are used to represent the volume of a cuboid.

Perimeter of a Cuboid

The cuboid’s perimeter is determined by its length, width, and height. Because the cuboid has 12 edges and the values of those edges varied, the perimeter is given by:

Perimeter of a Cuboid =  4 (l x b x h)

TSA of a Cube

The length, breadth and height of a cube are all equal. Let us assume the length of a cube as ‘a’. Hence,

Surface Area of a Cube = 2 (l x b + b x h + h x l) = 2 (a x a + a x a + a x a) = 2 (3a2) = 6a2

Surface Area of a Cube = 6a2

LSA of a Cube

The lateral area of a cube is the sum of the areas of the cube’s side faces. Because a cube has four side faces, its lateral area is the total of the areas of all four side faces.

Lateral Surface Area of a Cube = 4a2

Perimeter of a Cube

The perimeter of a cube is determined by the number of edges and the length of the edges. Because the cube has 12 edges and all of them are the same length, the perimeter of the cube is:

Perimeter of a Cube = 12a

What is a Cube?

A cube is a three-dimensional object which is formed when six identical squares bind to each other in an enclosed form. A cube has 6 faces, 12 edges, and 8 vertices. In other words, a cuboid whose length, breadth and height are equal is called a cube.

Cube

What is the Area of Square?

Volume of Cube

The formula to calculate the volume is length (l) × breadth (b) × height (h). Since l, b, h in a cube measure the same, its sides can be represented with a. So, l = b = h = a. Therefore,

volume of cube = l x b x h = a x a x a. Or simply,

Volume of Cube = a³

where a is the measurement of each side of the cube.

For example, the volume of the cube of side 1 cm will be equal to 1 cm × 1 cm × 1 cm = 1 cm³.

When a number is multiplied by itself three times, then the resulting number is called cube number. For example, 3 × 3 ×3  = 27. 27 is a cube number. Below is a list of cubes of first ten natural numbers. This list will come in handy for quick calculations.

Cubes of First 10 Natural Numbers

1³= 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1000

The numerical value obtained after cubing any given number is called a Perfect Cube. Knowing the properties of cube numbers will be helpful in calculating the volume of a cube.

Properties of Cube Number

  • Cubes of positive numbers are always positive. For example, cube of +4 is = (+4) × (+4) × (+4) = +64
  • Cubes of negative numbers are always negative. For example, cube of -4 is = (-4) × (-4) × (-4) = -64
  • Cubes of even numbers are always even.
  • Cubes of odd numbers are always odd.

Difference between Cube and Cuboid

  • All the edges (sides) of the cube are of equal lengths, but the edges of a cuboid are of different lengths.
  • All the sides of the cube are square in shape, whereas all the sides of the cuboid are rectangular in shape.
  • The area of all the faces of a cube are equal, but in a cuboid, only the area of the opposite faces are equal.
  • The diagonals of a cube are all equal, whereas only the diagonals of parallel sides of a cuboid are equal.

FAQs

Question 1. If six cubes of 10 cm edge are joined end to end, then the surface area (in sq. cm) of the resulting solid is

  1. 3600
  2. 3000
  3. 2600
  4. 2400

Answer: C. If six cubes are joined together, then the resulting solid is a cuboid. The dimensions of the solid are as follows,

length =60cm, breadth = 10 cm and height = 10 cm. Now, the total surface area of a cuboid is given by

2 [( l × b ) + ( b× h ) + ( l× h )]

By substituting the values, we get,
= 2 [( 60× 10 ) + ( 10× 10 ) + ( 60× 10)] = 2 ( 1300) = 2600 cm²

Question 2. John built a rectangular cardboard box 25 cm high with a square base and a volume of 2500 cm³. Then he realized he did not need a box that large, so he chopped off the height of the box reducing its volume to 1,000 cm³. Was the new box cubical? Also, state its’ height.
  1. 20 cm
  2. 10 cm
  3. 30 cm
  4. 40 cm

Answer: B. Volume of cuboid (V)  = length × breadth × height = Base area × height.

Given, V = 2500 cm3, height = 25 cm. By substituting the values in the formula, we get

Base area = 2500/25 = 100 cm²
Given that the base is a square. This means that the length = breadth. Therefore, the length of square base =  100 = 10 cm
After chopping, new volume = 1000 = 10 ×10 x new height
Therefore, new height = 1000/ 10 × 10 = 10 cm
As all the dimensions of the solid, l, b, h measure the same, the resulting solid is a cube.

Question 3: Find the surface area of a cube whose edge is 5 cm?

Answer: Cube is a type of cuboid in which the length, breadth and height measure the same. The formula to calculate the total surface area of a cuboid is 2 [( l × b ) + ( l× h ) + ( b× h )]. As l=b=h=a, the total surface area of cube= 2 [( a × a ) + ( a× a ) + ( a× a )] = 2 [3a2] = 6a2. To sum up,

Total surface area of cube = 6a2

Given that a = 5 cm. Thus,

T.S.A =6(52) = 6 X 25 = 150 cm2

Question 4: Define volume.

Answer: Volume is the amount of space that is occupied by a three-dimensional solid object. The unit of measurement of volume is the cubic meter.

Question 5: What is the lateral surface area of a cube of edge 10cm?

Answer: We know that the lateral surface area of a cuboid is given by 2(l+b)h. As a cube is a cuboid in which l=b=h=a, lateral surface area of cube = 2(a+a)a = 4a2. In simple terms

Lateral surface area of cube = 4a2

Given that a = 10 cm. Thus,

LSA = 4(102) = 400 cm2

Question 6: How do you measure the volume of water?

Answer: Basically, we can’t measure the volume of water until it is stored in a container, which can be a cube, cuboid, cylinder, cone, etc. And once it is inside a container we have to calculate the volume of the container to know the volume of water.

Question 7: Find the volume of a Rubik’s Cube of length 5 inches.

Answer: To find the volume of a Rubik’s Cube:

Given: Length of the side of a cube (a) = 5 in.

Using the Cube formula,

Volume = a x a x a = a3

Substituting the value of a,

Volume of a Rubik’s cube = (5)3 = 125 cubic inches

Share with friends
Customize your course in 30 seconds

Which class are you in?

5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started
Customize your course in 30 seconds

Which class are you in?

No thanks.