If a polyhedron has 9 edges and 6 vertices, how many faces does the solid have? A. 4 faces B. 6 faces C. 5 faces D. 7 faces Please select the best answer from the choices provided A B C D

The polyhedran has 5 faces (option C), if a polyhedron has 9 edges and 6 vertices. It is obtained by Euler's theorem.

Step-by-step explanation:

The given is,

                 Polyhedron  

                 9 edges

                 6 vertices

Step:1

                Formula to calculate No. of faces,

                               (1)

               The above equation is known as Polyhedral Formula or Euler's formula.

               Where, F - No. of faces

                            V - No. of vertices

                            E - No. of edges

              From the given,

                            V = 6

                            E = 9

             Equation (1) becomes,

                           F + 6 - 9 = 2

                                 F - 3 = 2

                                      F = 2 + 3

                 No. of face, F = 5

            The solid is Triangular prism, refer attachment.

Result:

         The polyhedran has 5 faces, if a polyhedron has 9 edges and 6 vertices. It is obtained by Euler's theorem.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

where are the options?

step-by-step explanation: