A regular polygon has 20 sides How many triangles can be drawn by using the vertices, but not using the sides?

Answer

To solve the problem of how many triangles can be formed using the vertices of a regular polygon with 20 sides, without using any of the polygon's sides, we can follow these steps:

### Step 1: Calculate the total number of triangles that can be formed using the vertices.
The total number of triangles that can be formed from 20 vertices is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of vertices and \( r \) is the number of vertices to choose (which is 3 for a triangle).

\[ \text{Total triangles} = \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140 \]

### Step 2: Subtract triangles that have at least one side in common with the polygon.
We need to consider two cases: triangles with two sides common and triangles with one side common.

#### Case 1: Triangles with two sides common
A triangle with two sides common to the polygon uses three consecutive vertices. Since there are 20 vertices, there are 20 such triangles (one for each set of three consecutive vertices).

\[ \text{Triangles with 2 sides common} = 20 \]

#### Case 2: Triangles with one side common
To form a triangle with one side common, we first choose one side of the polygon. There are 20 sides to choose from. After choosing a side (say vertices \( A \) and \( B \)), we cannot use the vertices \( A \) and \( B \) again, and we also cannot use the vertices adjacent to \( A \) and \( B \) (let's call them \( C \) and \( D \)). This leaves us with \( 20 - 4 = 16 \) vertices to choose from for the third vertex.

\[ \text{Triangles with 1 side common} = 20 \times 16 = 320 \]

### Step 3: Calculate the total number of triangles with at least one side common
Now we can add the triangles from both cases:

\[ \text{Total triangles with at least one side common} = 20 + 320 = 340 \]

### Step 4: Subtract the triangles with at least one side common from the total triangles
Finally, we subtract the triangles that have at least one side in common with the polygon from the total number of triangles:

\[ \text{Triangles without common sides} = 1140 - 340 = 800 \]

### Final Answer
The total number of triangles that can be drawn using the vertices of the polygon without using any of the polygon's sides is **800**.

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