**High School Number Sense Lesson 110: Creating Triangles from the Vertices of a Polygon**

This concept is similar to

**PolygonDiag**, which we covered late last year. It appeared

**7 times**this year on high school tests, with a median placement at

**question # 52**.

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There are two different variations of this problem:

- Finding the number of distinct triangles that can be drawn from
**one vertex**of a polygon - Finding the number of distinct triangles that can be drawn from
**any 3 vertices**of a polygon

**One vertex:**

- Subtract 2 from the number of sides of the polygon.
- Find that (n - 2)th
**triangular number**(**TriangularNum**).

**Three vertices:**

- Take the
**Combination**of the number of sides of the polygon, taken 3 at a time.

**Example 1: The number of triangles from a given vertex in a regular hexagon is ___**

- Subtract 2 from the number of sides: 6 - 2 =
**4**. - Find the
**4th**triangular number: 4(4 + 1)/2 = 4(5)/2 = 20/2 =**10**, which is your answer.

**Example 2: The number of triangles from a given vertex in a regular nonagon is ___**

- Subtract 2 from the number of sides: 9 - 2 =
**7**. - Find the
**7th**triangular number: 7(7 + 1)/2 = 7(8)/2 = 56/2 =**28**, which is your answer.

**Example 3: How many triangles can be formed using any three vertices of a regular pentagon? ___**

- Take the combination of the number of sides (5), taken 3 at a time.
- C(5,3) = 5!/(3!2!) = (5 x 4 x 3!)/(3!2!) = (5 x 4)/2! = 20/2 =
**10**.

**Example 4: The number of triangles formed from a given vertex in a regular octagon is ___**

- Subtract 2 from the number of sides: 8 - 2 =
**6**. - Find the
**6th**triangular number: 6(6 + 1)/2 = 6(7)/2 = 42/2 =**21**.

**Example 5: How many triangles can be formed using any three vertices of a regular dodecagon? ___**

- Take the combination of the number of sides (12), taken 3 at a time.
- C(12,3) = 12!/(3!9!) = (12 x 11 x 10 x 9!)/(3!9!) = (12 x 11 x 10)/3! = 1320/6 =
**220**.

**Up Next for High School: MatrixDeterm**