A polyhedron has 15 edges and 10 vertices. How many faces does it have?

1 Answer
Solace M. · Shwetank Mauria
Jun 8, 2017

#7# faces

Explanation:

In geometry, there is a really nifty, simple and extremely useful thing called Euler's formula, and it looks like this:
#V-E+F=2#, where
#V=#the number of vertices of a polyhedron
#E=#the number of edges of a polyhedron
#F=#the number of faces of a polyhedron.

A polyhedron is defined as a closed, solid object whose surface is made up of a number of polygonal faces (a polygon is defined as a plane figure with at least three straight sides and angles). A polyhedron can not have holes in it, and a polyhedron consists of one piece, so gluing one edge of a cube to another doesn't count as one.

This is where Euler's formula comes in. It tells us that for every (simple)polyhedron, the number of vertices minus the number of edges plus the number of faces will equal 2. Crazy right? Go ahead and try this formula out, it works!

Basically, this means we can solve your question using Euler's Formula, where
#V=10#
#E=15# and #F# is unknown. So lets substitute our known numbers into the formula to get

#10-15+F=2#

#-5+F=2#

From here we can add 5 to both sides to get

#F=7#, which means your shape has 7 sides.

It is a pentagonal prism.

enter image source here

I hope that helped!

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