Introduction

Cube and cuboid are three-dimensional shapes which consist of six faces, eight vertices and twelve edges. The primary difference between them is a cube has all its sides equal whereas the length, width and height of a cuboid are different. Both shapes look almost the same but have different properties. The area and volume of cube, cuboid and also cylinder differ from each other.
In everyday life, objects like a wooden box, a matchbox, a tea packet, a chalk box, a dice, a book, etc are encountered. All these objects have a similar shape. In fact, all these objects are made of six rectangular planes. In mathematics, the shape of these objects is either a cuboid or cube.
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CUBE

Definition :
A three-dimensional shape which has six square-shaped faces of equal size and has an angle of 90 degrees between them is called a cube.

->It has 6 faces, 12 edges and 8 vertices.
->Opposite edges are equal and parallel.
-> Each vertex meets three faces and three edges.

Sometimes, the shape of a cube is considered as “cubic”.

We can also say that a cube is considered as a block, where all the length, breadth and height are the same.

Along with that, it has 8 vertices and 12 edges such that 3 edges meet at one vertex point.

It is also known as a square parallelepiped, an equilateral cuboid and a right rhombohedron.

The cube is one of the platonic solids and it is considered as the convex polyhedron where all the faces are square.

We can say that the cube has octahedral or cubical symmetry.

A cube is the special case of the square prism.

In the above figure, you can see, edge, face and vertex of the cube.

Here, L stands for length, B stands for breadth and H stands for height.

We can see the length, breadth and height of the cube, which represents the edges of the cube, connected at a single point which is the vertex.

The faces of the cube are connected by four vertices.

Since the cube is a 3D shape, the two important parameters used to measure the cube are surface area and volume.

Properties of Cube:
The following are the important properties of cube:

->It has all its faces in a square shape.
->All the faces or sides have equal dimensions.
->The plane angles of the cube are the right angle.
->Each of the faces meets the other four faces.
->Each of the vertices meets the three faces and three edges.
->The edges opposite to each other are parallel.

CUBOID

Definition :

A three-dimensional figure which has three pairs of rectangular faces attached opposite to each other.
->These opposite faces are the same.
->Out of these six faces, two can be squares.
->The other names for cuboid are rectangular boxes, rectangular parallelepipeds, and right prisms.

Properties of a Cuboid :

Below are the properties of cuboid, its faces, base and lateral faces, edges and vertices.

Faces of Cuboid :

->A Cuboid is made up of six rectangles, each of the rectangles is called the face. In the figure above, ABFE, DAEH, DCGH, CBFG, ABCD and EFGH are the 6 faces of cuboid.
->The top face ABCD and bottom face EFGH form a pair of opposite faces. Similarly, ABFE, DCGH, and DAEH, CBFG are pairs of opposite faces. Any two faces other than the opposite faces are called adjacent faces.
->Consider a face ABCD, the adjacent face to this are ABFE, BCGF, CDHG, and ADHE.
Base and lateral faces :

->Any face of a cuboid may be called the base of the cuboid. The four faces which are adjacent to the base are called the lateral faces of the cuboid. Usually, the surface on which a solid rests on is known to be the base of the solid.
->In Figure above, EFGH represents the base of a cuboid.

Edges :

The edge of the cuboid is a line segment between any two adjacent vertices.
There are 12 edges, they are AB, AD, AE, HD, HE, HG, GF, GC, FE, FB, EF and CD and the opposite sides of a rectangle are equal.
Hence, AB=CD=GH=EF, AE=DH=BF=CG and EH=FG=AD=BC.

Vertices of Cuboid:

The point of intersection of the 3 edges of a cuboid is called the vertex of a cuboid.
A cuboid has 8 vertices A, B, C, D, E, F, G and H represents vertices of the cuboid in fig
By observation, the twelve edges of a cuboid can be grouped into three groups, such that all edges in one group are equal in length, so there are three distinct groups and the groups are named as length, breadth and height.
A solid having its length, breadth, height all to be equal in measurement is called a cube. A cube is a solid bounded by six square plane regions, where the side of the cube is called edge.

ACTIVITY-2

S.NO	Object	Length	Breadth	Height	Volume	Surface Area	Check
1
2
3

Important Formulas

Total surface area:
Cube= 6× (side)2
Cuboid= 2(lb + bh +lh)

Lateral surface area:
Cube= 4× (side)2
Cuboid= 2h(l +b)

Volume:
Cube= (side)3
Cuboid= (length × breadth × height)

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Relationship between Side Area and Volume

Is a cube, a special kind of cuboid?

Yes, a cube is a special kind of cuboid where all the faces of the cuboid are of equal length. In a cuboid, there are 6 faces which are rectangles. If the rectangles have equal sides, they become squares and eventually, the cuboid becomes a cube.

Calculating surface area and volume for cube and cuboid

Volume of CUBE

Enter the side of the cube: units

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Total Surface Area Of CUBE

Enter the side of the cube: units

Click the Calculate button to display the answer and Explanation for the problem explanation

Volume Of CUBOID

Enter the length of the cube : units
Enter the breadth of the cube : units
Enter the height of the cube : units

Click the Calculate button to display the answer and Explanation for the problem explanation

Total Surface Area Of CUBOID

Enter the length of the cube : units
Enter the breadth of the cube : units
Enter the height of the cube : units

Click the Calculate button to display the answer and Explanation for the problem explanation

Mentor: So, if we know that we can find the area of a two-dimensional figure,do you think that it is possible to find the area of a three dimensional figure? In fact, tell me what a three dimensional figure is?

Student: A three dimensional figure is like a ball or a cube--it's not flat.

Mentor: That's right. Now, can anyone say something about what it might mean to find the area of such a figure?

Student: When you say "find the area", do you mean the outside, or are the insides included?

Mentor: Well, it depends. There's actually no such thing as finding the "area" of a cube. Instead, we have the terms "volume" and "surface area". Let's talk about volume first. When you say "find the area" of a square, do you mean the outside, or are the insides included?

Student: We just look at the space it takes up on the paper; we assume the edges of the square have a width of zero.

Mentor: Precisely! Now imagine looking at the amount of space a cube takes up in 3 dimensions. We call that measure the volume of the cube.

Student: How do you measure the volume?

Mentor: Just as you measure and multiply the length and width of a rectangle to find its area, you multiply the length, width, and height of a 3-D object like a cube to find its volume. The three variables multiplied give it three dimensions, thus the volume, rather than simply the area. What do you suppose are the units for volume?

Student: Well, if there are three terms, all in inches, then it would be inches * inches * inches, which is inches cubed.

Mentor: How is this different from the units when you find area?

Student: Well, area is "squared" because you're just multiplying inches * inches.

Mentor: Exactly! Area is "squared" and volume is "cubed". How do you think that relates to their meaning?

Student: You find the area of a square or other two-dimensional objects, but you find the volume of three-dimensional objects like cubes!

Mentor: Good, we now know how to measure how much space an object takes up. But what about the outside of the object, as you mentioned earlier? What do you think "surface area" is?

Student: That sounds like it would be just the outsides--the area that is on the surface, which I can touch.

Mentor: Very well said! Surface area is the area of the surface of the three-dimensional shape. How would you calculate something like that?

Student: It seems almost too simple, but couldn't I just find the area of each two-dimensional face , then add the areas up?

Mentor: Absolutely! It is that simple. Almost all three dimensional objects you'll deal with are made up of two-dimensional faces that are just squares, triangles, etc, and the ones that are curved like spheres will have their own special formulas for surface area. Of course, the units for that are easy to find, right?

Student: Yep, it's just the standard units for area - units * units or units squared.

Mentor: You've got it! Now you're ready to try to solve some problems involving surface area and volume.