Can you imagine what this might look?

Geometry Level 2

A solid has 12 faces and 20 edges. Given that Euler's Formula applies, how many vertices does it have?

3 solutions

Sharky Kesa
May 23, 2014

Euler's Formula states that $f + v - 2 = e$ where $f$ is the number of faces, $v$ is the number of vertices and $e$ is the number of edges.

If we enter the known values into the equation we get

$12 + v - 2 = 20$

Solving the equation, we get the value of $v$ as 10.

12 faces but only 10 vertices? What in the world would the solid look like?

Marta Reece - 4 years, 6 months ago

Pentagonal anti-prism.

Sharky Kesa - 4 years, 6 months ago

Isn't it just 7 faces?

Saya Suka - 4 years, 5 months ago

@Saya Suka – Sorry, meant to write pentagonal anti-prism. Autocorrect must've done something dodgy.

Sharky Kesa - 4 years, 5 months ago

@Sharky Kesa – Did some digging and they said it is also called half isosceles icosahedron (2).

Saya Suka - 4 years, 5 months ago

The best I can imagine is 12 faces with 8 vertices.

Saya Suka - 4 years, 5 months ago

it will have 2 pentagonal faces and 10 triangular faces >v<

KHoa Nguyễn - 2 years, 5 months ago

In other words, a pentagonal anti-prism.

Sharky Kesa - 2 years, 5 months ago

Please explain in full with a diagram

Boadi Appiah-Adjei - 1 year, 6 months ago

A Former Brilliant Member
Jul 25, 2017

We can use the formula $V-E+F=2$ where $V$ is the number of vertices, $E$ is the number of edges and $F$ is the number of faces.

Entering the known quantities, we get

$V-20+12=2$

$V=2+20-12$

$\large \color{#D61F06}\boxed{V=10}$

Ryan Redz
May 28, 2014

V-E+F = 2, where E = 20, F = 12, then substitute, V - 20 + 12 = 2, so, V = 10