If the congruent polygons of a prism have 10 sides, how many faces, edges and vertices does the prism have? What if the polygons have sides?

Question

Homi · Accepted Answer

A prism is a 3D shape with two identical flat ends (called bases) and flat sides connecting them.

Step 1 — Identifying parts of a prism

Let us look at the diagrams of the prisms. We can see three main parts: faces, edges, and vertices.

Faces are the flat surfaces of the prism. Edges are the lines where two faces meet. Vertices are the points where three or more edges meet.

Diagram 1

Step 2 — Calculating for a 10-sided polygon prism

We are given that the congruent polygons (bases) of the prism have 10 sides. Let $s$ be the number of sides of the base polygon. So, $s = 10$ .

First, let us find the number of faces. A prism always has two base faces (one at the top and one at the bottom). It also has side faces, one for each side of the base polygon. Since the base has 10 sides, there will be 10 side faces.

Number of faces = Number of base faces + Number of side faces

$= 2 + s$

$= 2 + 10$

$\boxed{\text{Number of faces} = 12}$

Next, let us find the number of edges. Each base polygon has $s$ edges. Since there are two bases, that's $2 \times s$ edges for the bases. There are also $s$ edges connecting the vertices of the top base to the vertices of the bottom base. These are the vertical edges.

Number of edges = Edges on top base + Edges on bottom base + Connecting edges

$= s + s + s$

$= 3s$

$= 3 \times 10$

$\boxed{\text{Number of edges} = 30}$

Finally, let us find the number of vertices. Each base polygon has $s$ vertices. Since there are two bases, there will be $s$ vertices on the top base and $s$ vertices on the bottom base.

Number of vertices = Vertices on top base + Vertices on bottom base

$= s + s$

$= 2s$

$= 2 \times 10$

$\boxed{\text{Number of vertices} = 20}$

Step 3 — Generalizing for an n-sided polygon prism

Now, let us generalize these findings for a prism where the base polygon has $n$ sides. We will replace $s$ with $n$ in our formulas from Step 2.

Number of faces = Number of base faces + Number of side faces

$= 2 + n$

$\boxed{\text{Number of faces} = n + 2}$

Number of edges = Edges on top base + Edges on bottom base + Connecting edges

$= n + n + n$

$= 3n$

$\boxed{\text{Number of edges} = 3n}$

Number of vertices = Vertices on top base + Vertices on bottom base

$= n + n$

$= 2n$

$\boxed{\text{Number of vertices} = 2n}$

Answer

For a 10-sided polygon prism:

(i) Number of faces = 12 (ii) Number of edges = 30 (iii) Number of vertices = 20

For an n-sided polygon prism:

(i) Number of faces = n + 2 (ii) Number of edges = 3n (iii) Number of vertices = 2n

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