Polygon Sides: Spot The Non-Prime Length!

Hey guys! Ever wondered how geometry and prime numbers hang out? Well, buckle up because we're diving into a super cool problem that mixes polygons and primes. We've got a bunch of regular polygons with their perimeters, and the mission is to figure out which one doesn't have a side length that’s a prime number. Sounds like a fun brain workout, right? Let’s jump in!

Understanding the Problem

So, the main question we're tackling is this: Which regular polygon, with its given perimeter, does NOT have a side length that’s a prime number? To crack this, we need to remember a few key things. First off, what’s a regular polygon? It's simply a polygon where all the sides are the same length and all the angles are the same. Think of an equilateral triangle (3 sides), a square (4 sides), or a regular pentagon (5 sides)—all sides and angles are equal. Easy peasy!

Next up, perimeters. What's that all about? A polygon's perimeter is just the total distance around the outside—you get it by adding up the lengths of all the sides. So, if you've got a square with sides of 5 cm each, the perimeter is 5 cm * 4 sides = 20 cm. Got it? Great!

And last but not least, prime numbers. What makes a number prime? A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Numbers like 2, 3, 5, 7, and 11 are primes because you can’t divide them evenly by any other numbers except 1 and themselves. Now we’re armed with all the knowledge we need to solve this awesome puzzle.

Diving Deeper: The Connection Between Perimeters and Side Lengths

Before we leap into the specific polygons, let’s nail down how the perimeter and side length are related. This is the secret sauce to solving our problem! Since we’re dealing with regular polygons, every side is exactly the same length. This means the perimeter is simply the length of one side multiplied by the number of sides. Mathematically, we can write it like this:

Perimeter = Side Length * Number of Sides

Now, let's flip that around because we want to find the side length. To do that, we divide the perimeter by the number of sides:

Side Length = Perimeter / Number of Sides

This little formula is super important. It tells us that to find the side length, we just need to divide the total perimeter by how many sides the polygon has. With this in our toolkit, we’re ready to tackle each polygon and see if its side length is prime.

Understanding Prime Numbers: The Building Blocks of Our Solution

Alright, let's chat more about prime numbers because they’re the VIPs in our quest. We already know a prime number is a whole number greater than 1 that’s only divisible by 1 and itself. But why do they matter so much in this puzzle? Well, they’re like the basic building blocks of all other numbers. Any whole number can be made by multiplying prime numbers together—that’s pretty cool, huh?

Let's run through a few examples to make sure we’re on the same page. Take the number 7. Is it prime? Yep! The only numbers that divide evenly into 7 are 1 and 7. What about 12? Nope, 12 is divisible by 1, 2, 3, 4, 6, and 12. So, it’s not prime.

Now, think about even numbers. Are they prime? Well, most aren’t. Any even number greater than 2 can be divided by 2, so it has more than two divisors. That means 2 is the only even prime number. Keep this in mind—it’s a neat little shortcut that can save us time!

Analyzing the Polygons

Okay, let's get down to business and look at the polygons we’ve been given. We’re armed with the perimeters and the shapes, and now we just need to do some quick calculations to find the side lengths. Remember, we’re hunting for the polygon that doesn’t have a prime number for its side length. This means we’ll need to divide each perimeter by the number of sides for that polygon and see what we get.

A) Equilateral Triangle: Perimeter 39 cm

First up, the equilateral triangle. As we know, an equilateral triangle has 3 equal sides. The perimeter is 39 cm, so let's use our formula:

Side Length = Perimeter / Number of Sides Side Length = 39 cm / 3 Side Length = 13 cm

So, the side length of this equilateral triangle is 13 cm. Now, is 13 a prime number? You bet! The only numbers that divide evenly into 13 are 1 and 13. So, the equilateral triangle has a prime number side length.

B) Square: Perimeter 44 cm

Next, we have a square. Squares have 4 equal sides, and the perimeter is given as 44 cm. Let’s calculate the side length:

Side Length = Perimeter / Number of Sides Side Length = 44 cm / 4 Side Length = 11 cm

The side length of the square is 11 cm. Is 11 a prime number? Absolutely! Just like 13, 11 is only divisible by 1 and itself. So, the square also has a prime number side length.

C) Pentagon: Perimeter 85 cm

Now, let’s tackle the pentagon. A regular pentagon has 5 equal sides, and the perimeter is 85 cm. Time to find the side length:

Side Length = Perimeter / Number of Sides Side Length = 85 cm / 5 Side Length = 17 cm

This pentagon has a side length of 17 cm. Is 17 prime? Yes, indeed! Only 1 and 17 divide evenly into 17, so it’s another prime number. The pentagon joins the equilateral triangle and the square in having a prime number side length.

D) Another Polygon: Perimeter 54 cm

Last but not least, we have the final polygon with a perimeter of 54 cm. To figure out what kind of polygon it is, we need a little more info. However, the question already provides us with the side numbers of each polygon, so we can assume that this last one is a hexagon. A hexagon has 6 sides. Let’s calculate the side length:

Side Length = Perimeter / Number of Sides Side Length = 54 cm / 6 Side Length = 9 cm

So, the side length is 9 cm. Now, is 9 a prime number? Nope! 9 is divisible by 1, 3, and 9. Since it has more than two divisors, it’s not prime. Bingo! We’ve found our culprit.

Conclusion: The Non-Prime Side Length Champion

Alright, we’ve done the math, crunched the numbers, and taken a close look at each polygon. We found that the equilateral triangle, the square, and the pentagon all have side lengths that are prime numbers. But the final polygon, with a perimeter of 54 cm, has a side length of 9 cm, which is not a prime number. So, there you have it!

The polygon that does NOT have a prime number side length is the hexagon.

Isn't it amazing how math concepts like geometry and prime numbers can come together in such a neat problem? Hope you had fun working through this with me. Keep those brains buzzing and stay curious, guys!