Question

Apply Euler’s formula for a solid that has 16 faces and 21 edges, how many vertices does it have?
(a) 3
(b) 5
(c) 7
(d) 9

Answer 1

Hint: We will understand the concept of this Euler’s formula about who invented this formula and for what it is used for. Then we will apply the formula given as $V-E+F=2$ where V is vertices, E is called Edged and F is called faces. By substituting the values, in this we will get a number of vertices.

Complete step-by-step answer:
Euler’s formula tells us something very deep about shapes and space. This formula was named after the famous Swiss mathematician, Leonhard Euler. This formula is given as $V-E+F=2$ where V is vertices, E is called Edged and F is called faces. This little formula encapsulates a fundamental property of three-dimensional solids which are known as Polyhedra.
Now, a polyhedron is a solid object whose surface is made up of a number of flat faces which themselves are bordered by straight lines. Each face is in fact a polygon, a closed shape in the flat 2-dimensional plane made up of points joined by straight lines.
A polyhedron consists of polygonal faces, their sides are known as edges, and the corners as vertices. For example: Cube.

So, here we are given 16 faces and 21 edges. So, we will use the formula $V-E+F=2$ . On substituting the values, we will get
$V-21+16=2$
$V=2+5=7$
Thus, the number of vertices are 7.
Option (c) is the correct answer.


Note: Students should use this concept regarding Euler’s formula in order to solve this type of problems. Also, concepts about what is polyhedron should be known. Remember the formula properly otherwise mistake happens in writing the formula i.e. $V-F+E=2$ which will lead to the wrong answer. So, do not make this mistake.