Predict the number of diagonals in a heptagon
The correct answer and explanation is:
To calculate the number of diagonals in a heptagon, the formula for the number of diagonals in a polygon with nn sides is: Number of diagonals=n(n−3)2\text{Number of diagonals} = \frac{n(n – 3)}{2}
In the case of a heptagon, n=7n = 7. Substituting this value into the formula: Number of diagonals=7(7−3)2=7×42=282=14\text{Number of diagonals} = \frac{7(7 – 3)}{2} = \frac{7 \times 4}{2} = \frac{28}{2} = 14
Thus, a heptagon has 14 diagonals.
Explanation:
A diagonal is a line segment that connects two non-adjacent vertices in a polygon. In a heptagon, which has 7 sides and vertices, you can draw diagonals from each vertex to every other vertex except itself and its two adjacent vertices (since these do not form diagonals). This gives a total of n−3n – 3 diagonals per vertex, where nn is the number of sides or vertices.
Since there are 7 vertices in a heptagon, each vertex can form 7−3=47 – 3 = 4 diagonals. This gives a total of 7×4=287 \times 4 = 28 line segments. However, since each diagonal is counted twice (once from each endpoint), the total number of diagonals must be divided by 2, giving 282=14\frac{28}{2} = 14.
This method works for any polygon, not just a heptagon. The formula accounts for the fact that diagonals are not formed between adjacent vertices or between a vertex and itself, ensuring that the count remains accurate.