How many diagonals can be drawn by joining the vertices of an octagon ?

The correct answer is 20.

An octagon has 8 vertices. To find the number of diagonals, we use a systematic approach based on combinatorics.

Step 1: Count all possible line segments (connections) between any two vertices.

Any two vertices can be connected by a straight line. The number of ways to choose 2 vertices from 8 is given by the combination formula:

$$C^n{}_2=\frac{n!}{2!(n-2)!}$$

For an octagon, n = 8:

$$C^8{}_2=\frac{8!}{2!(8-2)!}=\frac{8\times 7}{2\times 1}=\frac{56}{2}=28$$

So there are 28 total line segments that can be drawn by joining any two vertices of the octagon.

Step 2: Subtract the sides of the octagon.

Not all 28 segments are diagonals. The segments connecting adjacent vertices form the sides of the octagon — and these are NOT diagonals. An octagon has exactly 8 sides.

Step 3: Calculate the number of diagonals.

$$\text{Number of Diagonals}=\text{Total Segments}-\text{Number of Sides}$$

$$\text{Number of Diagonals}=28-8=20$$

General Formula (for reference):
For any polygon with n sides, the number of diagonals is:

$$\text{Diagonals}=\frac{n(n-3)}{2}$$

For an octagon (n = 8):

$$\text{Diagonals}=\frac{8(8-3)}{2}=\frac{8\times 5}{2}=\frac{40}{2}=20$$

The formula $\frac{n(n-3)}{2}$ uses (n - 3) because from any vertex, you can draw a diagonal to every other vertex except itself and its 2 immediate neighbours (which form sides) — that leaves (n - 3) diagonals per vertex. Dividing by 2 avoids counting each diagonal twice.

Therefore, the number of diagonals that can be drawn in an octagon is 20. ✓