Problem 73
Question
Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of a polygon.) Pentagon
Step-by-Step Solution
Verified Answer
The pentagon has 5 diagonals.
1Step 1: Identify the given value
In this problem, the polygon is a pentagon which has 5 sides. So, the value of \(n\) is 5.
2Step 2: Substitute \(n\) in the Diagonal Formula
Substitute \(n=5\) in the formula for the number of diagonals \(\frac{n(n -3)}{2}\). This becomes \(\frac{5(5 - 3)}{2}\).
3Step 3: Simplify the equation
Simplify the equation to get the number of diagonals. \(\frac{5 \times 2}{2} = 5\).
Key Concepts
Understanding a PentagonExplaining the Diagonal FormulaExploring Polygon Properties
Understanding a Pentagon
A pentagon is a five-sided polygon. It is a simple geometric shape that can be regular or irregular.
In a regular pentagon, all sides and angles are equal. This symmetry gives it a balanced and attractive appearance. However, pentagons can also be irregular, meaning not all sides or angles are the same.
Pentagons can be found in many places, such as in the structure of certain flowers, architecture, and even in popular logos. Their five-sided nature makes them unique and an interesting subject when studying polygons.
In a regular pentagon, all sides and angles are equal. This symmetry gives it a balanced and attractive appearance. However, pentagons can also be irregular, meaning not all sides or angles are the same.
Pentagons can be found in many places, such as in the structure of certain flowers, architecture, and even in popular logos. Their five-sided nature makes them unique and an interesting subject when studying polygons.
Explaining the Diagonal Formula
The diagonal formula is a useful mathematical tool to find the number of diagonals in a polygon. A diagonal is a line connecting two non-adjacent vertices in a polygon. For a polygon with sides, the formula to find the number of diagonals is \[ \frac{n(n-3)}{2} \].
The formula works by calculating all possible connections between vertices, which is \( \frac{n(n-1)}{2} \), and then subtracting the lines that are actually the polygon's sides, which is \(n\) itself. Hence, the simplified form is \( \frac{n(n-3)}{2} \).
This formula is derived considering that each vertex connects to \(n-1\) other vertices, and since the connections include the sides of the polygon, we must remove these side connections in our calculation. By dividing by two, we also ensure that no diagonal is counted more than once.
The formula works by calculating all possible connections between vertices, which is \( \frac{n(n-1)}{2} \), and then subtracting the lines that are actually the polygon's sides, which is \(n\) itself. Hence, the simplified form is \( \frac{n(n-3)}{2} \).
This formula is derived considering that each vertex connects to \(n-1\) other vertices, and since the connections include the sides of the polygon, we must remove these side connections in our calculation. By dividing by two, we also ensure that no diagonal is counted more than once.
Exploring Polygon Properties
Polygons are essential shapes in geometry, made up of straight sides. They can have any number of sides larger than three.
Here are a few essential properties of polygons:
Here are a few essential properties of polygons:
- Sides: The number of sides, \(n\), determines the classification of the polygon (triangle, quadrilateral, pentagon, etc.).
- Angles: The sum of interior angles for any polygon is \((n-2) \times 180^{\circ}\).
- Diagonals: The number of diagonals can be calculated using the diagonal formula.
Other exercises in this chapter
Problem 72
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