Formula for the Number of Diagonals

Oct 13, 2025 by leolavanderia.com.br 43 views

Introduction

Diagonals are line segments that connect two non-adjacent vertices of a polygon. The number of diagonals in a polygon is a fundamental concept in geometry, and it has numerous applications in mathematics and other fields. In this article, we will delve into the world of polygons and explore the relationship between the number of diagonals and the number of sides of a polygon.

What are Diagonals?

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. In other words, it is a line segment that connects two vertices that are not next to each other. For example, in a square, the line segments connecting the opposite corners are diagonals.

Formula for the Number of Diagonals

The number of diagonals in a polygon can be calculated using a simple formula. The formula is as follows:

Number of diagonals = n(n-3)/2

where n is the number of sides of the polygon.

Derivation of the Formula

To derive the formula, let's consider a polygon with n sides. Each vertex can be connected to (n-3) other vertices using diagonals. However, this counts each diagonal twice, once for each vertex it connects. Therefore, we need to divide the total count by 2 to get the correct number of diagonals.

Applying the Formula

Now that we have the formula, let's apply it to the given problem. We are given that the number of diagonals is 65. We can set up the equation as follows:

n(n-3)/2 = 65

Solving the Equation

To solve the equation, we can start by multiplying both sides by 2 to eliminate the fraction:

n(n-3) = 130

Expanding the Equation

Next, we can expand the equation by multiplying the two terms:

n^2 - 3n = 130

Rearranging the Equation

Now, we can rearrange the equation to form a quadratic equation:

n^2 - 3n - 130 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -3, and c = -130.

Calculating the Solutions

Plugging in the values, we get:

n = (3 ± √((-3)^2 - 4(1)(-130))) / 2(1) n = (3 ± √(9 + 520)) / 2 n = (3 ± √529) / 2 n = (3 ± 23) / 2

Finding the Positive Solution

Since the number of sides cannot be negative, we take the positive solution:

n = (3 + 23) / 2 n = 26 / 2 n = 13

Conclusion

Therefore, the number of sides of the polygon is 13.

Final Answer

The final answer is c) 13.

Introduction

In our previous article, we explored the relationship between the number of diagonals and the number of sides of a polygon. We derived a formula to calculate the number of diagonals and applied it to a specific problem. In this article, we will provide a Q&A guide to help you understand the concept better.

Q1: What is a diagonal in a polygon?

A1: A diagonal is a line segment that connects two non-adjacent vertices of a polygon.

Q2: How many diagonals can be drawn from a single vertex in a polygon?

A2: Each vertex can be connected to (n-3) other vertices using diagonals, where n is the number of sides of the polygon.

Q3: Why do we need to divide the total count of diagonals by 2?

A3: We need to divide the total count by 2 because each diagonal is counted twice, once for each vertex it connects.

Q4: What is the formula for the number of diagonals in a polygon?

A4: The formula is n(n-3)/2, where n is the number of sides of the polygon.

Q5: How do we apply the formula to find the number of diagonals?

A5: We can apply the formula by plugging in the value of n into the equation and solving for the number of diagonals.

Q6: What is the significance of the number of diagonals in a polygon?

A6: The number of diagonals in a polygon is an important concept in geometry and has numerous applications in mathematics and other fields.

Q7: Can we find the number of sides of a polygon if we know the number of diagonals?

A7: Yes, we can find the number of sides of a polygon if we know the number of diagonals by using the formula and solving for n.

Q8: How do we solve the quadratic equation that arises from the formula?

A8: We can solve the quadratic equation by using the quadratic formula: n = (-b ± √(b^2 - 4ac)) / 2a.

Q9: What is the final answer to the problem of finding the number of sides of a polygon with 65 diagonals?

A9: The final answer is 13.

Q10: Can you provide an example of a polygon with 13 sides?

A10: Yes, a polygon with 13 sides is a tridecagon.

Conclusion

In this Q&A guide, we have provided answers to common questions related to the number of diagonals in a polygon. We have also provided examples and explanations to help you understand the concept better.

Final Tips

Make sure to understand the formula and how to apply it to find the number of diagonals.
Practice solving quadratic equations to find the number of sides of a polygon.
Use the formula to find the number of diagonals in different polygons.

Additional Resources

For more information on geometry and polygons, visit the following websites:

Khan Academy: Geometry
Math Open Reference: Polygons
Wolfram MathWorld: Polygons

Final Answer

The final answer is c) 13.