How many triangles can be formed by joining the vertices of an octagon? a. 52 b. 56 c. 72 d. 44
The Correct Answer and Explanation is:
Correct Answer: b. 56
To determine how many triangles can be formed by joining the vertices of a regular octagon, it is important to consider the total number of ways to choose any 3 vertices from the 8 vertices of the octagon.
The number of combinations of 3 vertices from 8 is calculated using the combination formula:(83)=8×7×63×2×1=56\binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56(38)=3×2×18×7×6=56
This result represents all possible sets of 3 non-collinear points that can form a triangle. In a regular polygon such as an octagon, no three vertices lie on the same straight line, ensuring that every set of 3 vertices forms a triangle.
Each triangle formed is unique based on the different combinations of vertices. These triangles can be classified based on their positions, such as those fully inside the octagon or those that use edges of the polygon, but the total count remains the same regardless of position or orientation.
It is also worth noting that this calculation includes all types of triangles—scalene, isosceles, and possibly equilateral (although regular polygons with more than 6 sides do not produce equilateral triangles using only their vertices).
In conclusion, choosing any 3 out of 8 distinct vertices on the octagon will always yield a triangle, and the total number of such combinations is 56. Thus, the correct choice is b. 56.
