A regular dodecahedron has 30 edges. How many vertices does this solid have?

Introduction / Context:
This question is about a regular dodecahedron, which is one of the five Platonic solids. It is a three dimensional solid with congruent regular pentagonal faces. The problem gives the number of edges and asks for the number of vertices. To answer this, we use Euler's formula for convex polyhedra, which relates the number of faces, vertices, and edges. This is an important concept in solid geometry and topology, and it appears frequently in aptitude and competitive exams.

Given Data / Assumptions:
• The solid is a regular dodecahedron.
• It is known that a regular dodecahedron has 12 faces, each a regular pentagon.
• The number of edges, E, is given as 30.
• We are asked to find the number of vertices, V.
• The solid is convex, so Euler's polyhedron formula applies: V - E + F = 2.

Concept / Approach:
Euler's formula for any convex polyhedron states that V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. For a dodecahedron, F is known to be 12, and E is given as 30. Substituting these values into Euler's formula allows us to solve for V directly. This approach avoids having to memorize all properties of a dodecahedron and instead relies on a general relationship that works for many solids.

Step-by-Step Solution:
Step 1: Write Euler's formula for a convex polyhedron: V - E + F = 2. Step 2: For a regular dodecahedron, the number of faces F is 12. Step 3: The number of edges E is given as 30. Step 4: Substitute E = 30 and F = 12 into Euler's formula: V - 30 + 12 = 2. Step 5: Simplify: V - 18 = 2. Step 6: Solve for V: V = 2 + 18 = 20.

Verification / Alternative check:
We can also verify this result using the relationship among faces, edges, and vertices of regular polygons forming the solid. Each pentagonal face has 5 edges, and each edge is shared by 2 faces. Thus, the total edge count satisfies 5F = 2E, so 5 * 12 = 2E, giving E = 30, which is consistent. Also, each vertex of a regular dodecahedron is where 3 pentagons meet. Counting incidences of vertices across faces gives 5F = 3V, so 60 = 3V, giving V = 20. This matches the value from Euler's formula and confirms the answer.

Why Other Options Are Wrong:
10 and 12 are too small and would not satisfy Euler's formula with E = 30 and F = 12; substituting these numbers leads to V - E + F values far from 2.
22 and 24 are too large; they also break the relation 5F = 3V for regular pentagonal faces meeting three at a vertex. None of these satisfy both Euler's formula and the face structure of a dodecahedron.

Common Pitfalls:
Some learners confuse the dodecahedron with other Platonic solids such as the icosahedron or cube and may guess vertex counts based on memory rather than using Euler's formula. Others misapply Euler's formula or misremember the number of faces of a dodecahedron. To avoid such errors, it is safer to recall that a dodecahedron has 12 pentagonal faces and then rely on V - E + F = 2 and 5F = 2E or 5F = 3V for regular structure to verify counts.

Final Answer:
A regular dodecahedron with 30 edges has 20 vertices.