Euler's Formula for 3 Dimensional Shapes Video Lecture - Class 8

Is there a relationship between the faces,
vertices and edges of a straight face solid?
Every straight face solids has faces, vertices and edges.
Take the example of a cube.
This is how it's three dimensional view looks like.
We need to find the number of faces,
the number of vertices and the number of edges in a cube.
Let's count the number of faces first.
It has six faces in all.
Top and bottom, front and back, left and right.
We write a 6 here.
How many vertices does it have?
Four at the top and four at the bottom, eight in all.
And how many edges?
Pause the video and write the number of edges here.
Let's count!
Four horizontal ones at the top,
four vertical ones in the middle
and four horizontal ones at the bottom.
12 edges in total.
Now here's the important question.
Is there a relation between these three parameters
of a straight face solid?
The answer is yes!
For any straight face solid
if we add its number of faces to its number of vertices
and then subtract the number of edges,
we get one special number! And that number is two.
For a cube '6 plus 8 minus 12' gives us '2'.
And this holds true for any straight face solid.
Which are the other straight face solids we know of?
Cuboid is 1, it looks like this.
A cube is nothing but a special case of a cuboid where the length,
breadth and the height are equal.
So a cuboid also has 6 faces.
8 vertices and 12 edges.
And 'F plus V minus E' will give us 2.
Let's see if it works for other straight face solids.
Do you remember what a square pyramid looks like?
It is a figure with a square as its base and triangles as its faces.
Square base and 4 triangles as faces.
And what is a triangular prism?
A prism is a 3 dimensional solid which has identical and parallel ends
and all flat faces.
A triangular prism would look like this.
It has two identical and parallel triangles at its ends
and three rectangular faces.
What do we need to find out here?
We need to find the number of faces
the number of vertices
and the number of edges in each of the solids
and see if 'F plus V minus E' will give us 2.
Okay, let's tackle the first solid a square pyramid.
How many faces do you see?
There are four triangular faces and one square face.
Five in all.
How many vertices?
One at the top and four at the base!
Five vertices. And how many edges?
Four slant edges and four edges at the base.
A total of eight edges.
We got F, V and E as 5, 5 and 8 respectively.
And '5 plus 5 minus 8' will give us 2.
So it worked for the square pyramid too.
'F plus V minus E' gave us '2'.
This is called ''Euler's formula''.
For any straight face solid
'F plus V minus E' will always give us '2'.
Will Euler’s formula work for the triangular prism too ?
Why don't you give it a try?
Okay, so how many faces?
Two triangular faces at the ends
and three rectangular faces in between.
Five faces.
How many vertices?
Three at this end and three at the other, six vertices.
Three edges at the front face
three in between and three at the back face!
Nine edges in all.
What will be '5 plus 6 minus 9'?
Yes, we get a 2.
We will always get a 2.
That's Euler's formula.
‘F plus V minus E’ will always give us a '2'.