Calculate Cone Height: Radius 5cm, Volume 100pi

Dec 11, 2025 by Andrew McMorgan 48 views

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically tackling a classic problem: finding the height of a cone. You know, those cool pointy things that look like party hats or ice cream cones? We've got a specific challenge for you today: a cone with a radius of 5 cm and a volume of 100π cubic cm. Sounds like a mouthful, right? But don't sweat it! By the end of this article, you'll be a pro at figuring out that missing height, and you'll understand the magic behind the formula. We're going to break it down step-by-step, making it super easy to follow. So, grab your notebooks, maybe a slice of pizza (since we're talking cones!), and let's get this math party started!

Understanding the Cone Volume Formula

Alright team, before we can find the height, we absolutely need to get cozy with the volume of a cone formula. This is like the secret handshake for solving any cone-related problem. The formula is:

$V = \frac{1}{3} \pi r^2 h$

Let's break down what each of these awesome letters means, shall we?

V stands for the Volume of the cone. This is the total space inside the cone, measured in cubic units (like cubic centimeters, or cm³).
π (pi) is that famous mathematical constant. You know, the one that's approximately 3.14159. It pops up everywhere in circle and sphere related stuff!
r represents the radius of the cone's base. The base is that perfectly round circle at the bottom of the cone. The radius is the distance from the center of that circle to its edge.
h is what we're trying to find – the height of the cone! This is the perpendicular distance from the apex (the pointy tip) straight down to the center of the base. It's like how tall the cone stands up.

So, what this formula is basically telling us is that the volume of a cone is one-third of the volume of a cylinder with the same base radius and height. Pretty neat, huh? It's like the cone is a more 'slender' version of its cylindrical cousin. To solve our problem, we'll be rearranging this formula to isolate 'h', our mysterious height. Don't worry, it's not as scary as it sounds! We'll walk through the algebra together, making sure everyone's on board. This formula is your golden ticket, guys, so make sure you've got it memorized or written down somewhere handy!

Plugging in the Known Values

Okay, squad, we've got our trusty formula, and we know our mission: find the height. Now, let's see what awesome information the problem has already given us. We are told that the radius (r) is 5 cm and the volume (V) is 100π cm³. This is fantastic news! It means we don't have to guess; we have concrete numbers to work with. So, let's take our volume formula, $V = \frac{1}{3} \pi r^2 h$ , and start swapping in the values we know.

Where we see 'V', we'll put '100π'. And where we see 'r', we'll replace it with '5'. So, our equation now looks like this:

$100\pi = \frac{1}{3} \pi (5)^2 h$

See? We're already making progress! It’s like filling in the blanks on a cool math puzzle. The $(5)^2$ part just means 5 multiplied by itself, which is 25. So, we can simplify that part of the equation.

$100\pi = \frac{1}{3} \pi (25) h$

And we can rearrange the terms on the right side a little bit to make it look cleaner:

$100\pi = \frac{25\pi}{3} h$

Now, this equation has all the information we need to solve for 'h'. We've successfully translated the word problem into a mathematical equation that's ready for us to work our algebraic magic on. It's crucial at this stage to be super careful with your numbers and symbols. One tiny slip-up can lead you down the wrong path, and we don't want that! We're almost there, guys. Just a few more steps to uncover the secret height of our cone!

Solving for the Height (h)

Alright mathletes, we've reached the critical point: solving for the height (h). We've got our equation: $100\pi = \frac{25\pi}{3} h$ . Our goal here is to get 'h' all by itself on one side of the equation. Think of it like isolating your favorite video game controller – you want it all to yourself! To do this, we need to undo the operations that are being done to 'h'. Currently, 'h' is being multiplied by $\frac{25\pi}{3}$ . To get rid of that fraction and the pi, we can do a couple of smart moves.

First, let's tackle the fraction $\frac{1}{3}$ that's multiplying the $25\pi h$ . The easiest way to get rid of the denominator (the '3' at the bottom) is to multiply both sides of the equation by 3. This keeps our equation balanced, like a perfectly weighted seesaw.

$3 * (100\pi) = 3 * (\frac{25\pi}{3} h)$

This simplifies to:

$300\pi = 25\pi h$

Now, 'h' is being multiplied by $25\pi$ . To isolate 'h', we need to divide both sides of the equation by $25\pi$ . This cancels out the $25\pi$ on the right side, leaving 'h' all alone.

$\frac{300\pi}{25\pi} = \frac{25\pi h}{25\pi}$

On the left side, the 'π' on the top and bottom cancel each other out, which is super convenient! We're left with $\frac{300}{25}$ . Now, we just need to perform that division. Think of it like this: how many quarters (25 cents) are in $3.00? Or, how many 25s fit into 300?

$300 \div 25 = 12$

And on the right side, the $25\pi$ on the top and bottom cancel out, leaving us with just 'h'.

So, we have:

$12 = h$

Boom! Just like that, we've found our answer. The height of the cone is 12 cm. It's that simple when you break it down. Remember, algebra is all about using inverse operations to isolate the variable you're looking for. Keep practicing these steps, and you'll be solving for unknowns like a boss in no time!

Final Answer and Recap

So, there you have it, my friends! After a bit of mathematical detective work, we have successfully determined the height of the cone. We started with a cone that had a radius of 5 cm and a volume of 100π cm³, and through the power of the volume formula, we found that its height is exactly 12 cm. How awesome is that?

Let's quickly recap the journey we took:

We recalled the fundamental formula for the volume of a cone: $V = \frac{1}{3} \pi r^2 h$ . This is our blueprint for all cone volume calculations.
We identified the given information: Radius ( $r = 5$ cm) and Volume ( $V = 100\pi$ cm³).
We substituted these known values into the formula: $100\pi = \frac{1}{3} \pi (5)^2 h$ .
We simplified the equation: $100\pi = \frac{25\pi}{3} h$ .
We used algebraic manipulation (multiplying by 3 and dividing by $25\pi$ ) to isolate the height ( $h$ ).
We arrived at our final answer: $h = 12$ cm.

This process isn't just about finding a number; it's about understanding the relationships between different properties of a geometric shape. Whether you're designing a cool new product, building a structure, or just flexing your brain muscles with some geometry, these skills are super valuable. Don't be afraid to tackle more problems like this. The more you practice, the more confident you'll become. Keep exploring the fascinating world of mathematics, and remember, every problem is just an opportunity to learn something new and awesome! Catch you in the next one, guys!