Find Polygon Sides With Interior Angle Sum

Ever wondered how many sides a polygon has just by knowing the sum of its interior angles? It's a common question in geometry, and thankfully, there's a neat formula to help us figure it out! The formula, s = 180(n-2), is your key. Here, 's' represents the sum of the interior angles, and 'n' stands for the number of sides of the polygon. We're going to dive deep into understanding this formula and then apply it to a specific problem: finding the number of sides of a polygon when the sum of its interior angles is 1,260°.

Understanding the Polygon Interior Angle Formula

Let's break down the formula s = 180(n-2). This equation is fundamental in understanding the properties of polygons. The number of sides ('n') dictates the shape and the sum of its internal angles. Imagine a triangle (n=3). The sum of its interior angles is always 180°. If you plug n=3 into the formula, you get 180(3-2) = 180(1) = 180°. See? It works! Now, consider a quadrilateral, like a square or a rectangle (n=4). The sum of its interior angles is 360°. Using the formula: 180(4-2) = 180(2) = 360°. It holds true! This formula is derived from the idea that any polygon can be divided into a certain number of triangles. For an n-sided polygon, you can always draw (n-2) non-overlapping triangles from one vertex to all other non-adjacent vertices. Since each triangle has an interior angle sum of 180°, the total sum for the polygon is simply the number of triangles multiplied by 180°. This is why the formula is so elegant and powerful. It links the number of sides directly to the total measure of the internal angles, providing a consistent rule for all convex polygons. The beauty of this formula lies in its simplicity and its universal applicability across different types of polygons. Whether it's a simple square or a complex decagon, this mathematical relationship remains unchanged. It's a cornerstone of Euclidean geometry, allowing us to calculate and verify angle properties with ease. Understanding why this formula works, by visualizing the triangulation of polygons, solidifies its concept and makes it easier to remember and apply in various geometric problems. We'll use this core understanding to solve our specific problem.

Solving the Polygon Problem

Now, let's tackle the specific problem: how many sides does a polygon have if the sum of the interior angles is 1,260°? We have our trusty formula: s = 180(n-2). We know that 's' is 1,260°. So, we can plug that value into the equation: 1,260 = 180(n-2). Our goal is to solve for 'n', the number of sides. First, we need to isolate the term with 'n' in it. We can do this by dividing both sides of the equation by 180. So, 1,260 divided by 180 equals (n-2). Let's do the division: 1260 / 180. You can simplify this by canceling out a zero from both numbers, making it 126 / 18. If you know your multiplication tables, you'll recognize that 18 * 7 = 126. So, 1260 / 180 = 7. Now our equation looks like this: 7 = n - 2. To find 'n', we simply need to add 2 to both sides of the equation. 7 + 2 = n. Therefore, n = 9. So, a polygon with an interior angle sum of 1,260° has 9 sides. This means it's a nonagon! It's amazing how a simple algebraic manipulation of a geometric formula can reveal such specific information about a shape. This process highlights the interconnectedness of different mathematical concepts, showing how algebra is a powerful tool for solving geometric puzzles. The steps are clear: substitute the known value, perform the necessary arithmetic operations (division and addition in this case), and arrive at the unknown variable, which represents the number of sides. This method is robust and can be applied to any polygon where the sum of the interior angles is provided. The key is to correctly apply the formula and solve the resulting linear equation. This practical application of the polygon angle sum theorem is a great way to reinforce your understanding of both geometry and basic algebra. It's a satisfying process that leads to a definitive answer about the polygon's structure.

Connecting to the Options

We found that the polygon has 9 sides. Let's look at the options provided:

A. 6 sides B. 7 sides

It seems there might be a slight discrepancy between our calculated answer and the options given. Let's re-check our calculation carefully to ensure accuracy. We started with the formula s = 180(n-2) and the given sum of interior angles s = 1,260°. Substituting the value of 's', we get 1,260 = 180(n-2). To solve for 'n', we first divide both sides by 180:

1260 / 180 = n - 2

Performing the division: 1260 / 180 = 7.

So, the equation becomes 7 = n - 2.

Now, we add 2 to both sides to find 'n':

7 + 2 = n

n = 9.

Our calculation consistently shows that the polygon has 9 sides. It's possible that the provided options might have been intended for a different sum of angles, or there might be a typo in the options themselves. For example, if the sum of the interior angles was 900°, then 900 = 180(n-2), which simplifies to 5 = n-2, and thus n = 7. This would match option B. If the sum was 720°, then 720 = 180(n-2), which simplifies to 4 = n-2, and thus n = 6. This would match option A. However, based on the given sum of 1,260°, the correct answer is indeed 9 sides. This is a good reminder that in mathematical problems, it's crucial to trust your calculations, but also to be aware that errors can sometimes occur in the problem statement or the provided choices. If this were a test, and you were confident in your work, you might note the discrepancy. For the purpose of this explanation, we have confirmed that a polygon with an interior angle sum of 1,260° has 9 sides.

Conclusion

The formula s = 180(n-2) is an indispensable tool for anyone studying geometry. It elegantly connects the sum of a polygon's interior angles to its number of sides. By understanding this formula, we can solve a variety of problems, including determining the number of sides when the angle sum is known. In our case, with an interior angle sum of 1,260°, we algebraically manipulated the formula to find that the polygon must have 9 sides. While this didn't perfectly align with the given options, our step-by-step derivation confirms the result. This exercise highlights the power of applying mathematical principles consistently. For further exploration into the fascinating world of polygons and their properties, you might find resources from Khan Academy to be incredibly helpful and insightful.