Class 8 Maths Visualising Solid Shapes Euler’s formula

Euler’s formula

For any polyhedron,

F + V – E = 2

where ‘F’ stands for number of faces, V stands for number of vertices and E stands for number of edges.

This relationship is called Euler’s formula.

Problem: Can a polygon have for its faces:

(i) 3 triangles                  (ii) 4 triangles                  (iii) a square and four triangles

Solution:

(i) No, a polyhedron cannot have 3 triangles for its faces.

(ii) Yes, a polyhedron can have four triangles which is known as pyramid on triangular base.

(iii) Yes, a polyhedron has its faces a square and four triangles which makes a pyramid on square base.

Problem: Verify Euler’s formula for these solids. Solution:

(i) Here, figure (i) contains 7 faces, 10 vertices and 15 edges.

Using Euler’s formula, we see F + V – E = 2

Putting F = 7, V = 10 and E = 15,

F + V – E = 2

=> 7 + 10 – 5 = 2

=> 17 – 15 = 2

=> 2 = 2

=> L.H.S. = R.H.S.

(ii) Here, figure (ii) contains 9 faces, 9 vertices and 16 edges.

Using Euler’s formula, we see F + V – E = 2

F + V – E = 2

=> 9 + 9 – 16 = 2

=> 18 – 16 = 2

=> 2 = 2

=> L.H.S. = R.H.S.

.