What is volume?

Welcome to Maths Is Not Boring by kidopoly.com. I'm Mira! And I'm Finn! Before we start, a big hello to August, Oliver and Archie in Bixby, Oklahoma, who listen together every single day on the way to school.

August, Oliver and Archie. The strange thing we are exploring today is something you might see right out your car window on your way to school. Or even in your cereal bowl.

And if you haven't already, tap follow wherever you're listening so you never miss an episode.

Okay, Mira. What are we looking at today?

Well, Finn, imagine you have a perfectly smooth, straight building block. It is pretty easy to measure how much space it takes up, right? You just use a ruler on the straight edges.

Oh yeah, you just measure the bottom, the sides, and the top. Easy math.

Exactly. But now, imagine you are holding a lumpy, twisty, bumpy plastic dragon toy. It has tiny sharp claws, a long curved tail, and weird little gaps between its teeth.

Wait, you can't use a ruler on a dragon's teeth. The shape is way too crazy. There are too many curves and spikes.

Exactly! How do you measure exactly how much space that strange impossible shape takes up? Today, we are talking about a mathematical idea that solves that impossible problem. And it all started with a bath. So, the idea we are looking at is called volume. Volume is just the mathematical name for exactly how much room a thing takes up in the universe.

How much room it takes up in 3D space. Like how much air fits inside a balloon, or how much stuff is packed into a baseball.

Spot on. For thousands of years, measuring the volume of neat, flat things was simple. But measuring the volume of lumpy, uneven things seemed totally impossible. Until we get to a man named Archimedes. Archimedes? When did he live? He lived in a beautiful ancient city called Syracuse on the island of Sicily, about 250 years before the Year One.

So around 250 BCE.

That is over 2,000 years ago. They didn't have computers or lasers to measure stuff.

Nope, but Archimedes was basically the smartest problem solver in the ancient world. The ruler of Syracuse was King Hyro, and the king had a very stressful problem. What kind of problem? Did he lose something important? Worse, he thought he was being scammed. The king had given a goldsmith a heavy block of pure, solid gold, and ordered him to make a magnificent royal crown.

Ooh, kings love a big shiny crown.

The goldsmith brought back a stunning crown, shaped like twisting leaves and delicate branches. The king weighed it, and it weighed exactly the same as the block of gold he had originally handed over.

Okay, so what is the problem? The weight was totally perfect.

The king was paranoid. He worried the sneaky goldsmith had kept some of the super expensive gold for himself and swapped in some cheaper silver to make the weight perfectly match. Oh, a royal trick!

But how could the king ever prove it if the scale said the weight was exactly right? Exactly.

The king ordered Archimedes to find the truth, but there was a catch. Archimedes was absolutely not allowed to melt the beautiful crown down, or scratch it, or damage it in any way.

Wait, hold on. If silver and gold weigh the same amount on a scale, how does mixing in silver change anything?

That is the secret of volume. Think of it like a pound of rocks versus a pound of feathers. They both weigh exactly one pound.

Oh, but to get a pound of feathers, you need a massive sack of them. The rocks just fill a tiny bag. Yes.

Pure gold is incredibly heavy for its size. It packs a lot of weight into a tiny space, but silver is lighter. To get the same weight in silver, you need to use a lot more of it. It is bulkier.

So if there was silver hidden in the crown, the crown would have to be slightly bigger. It would take up a tiny bit more space than pure gold.

Exactly. It would have more volume. If the leafy crown had a slightly bigger volume than a block of pure gold that weighed the same, the king was definitely being cheated.

But we are back to the dragon toy problem. The crown is full of twisty metal leaves and weird gaps. Nobody can measure the volume of a twisty branch with a straight ruler.

That is exactly what Archimedes was agonizing over. How do you measure the exact volume of a messy, complicated shape? And then he went to take a bath.

A bath to wash his worries away.

He stepped into a bathtub that was filled right to the very brim. And as he lowered his body into the water, the water spilled over the sides onto the floor.

Well, yeah. His body was taking up space in the tub. The water couldn't squish down, so it had to go somewhere.

Finn, you just saw what Archimedes saw. The water didn't just splash, it was pushed out of the way. The water perfectly wrapped around every curve of his body, measuring him exactly. This incredible idea is called displacement. A submerged object pushes aside exactly its own volume of water. The water acts like a perfect, flexible 3D mould.

It does not matter how weird or twisty the shape is.

That is genius! So, to find the volume of the crazy twisty crown, you just drop it in a full bowl of water and measure exactly how much water spills out over the edge.

Yes! The legend says Archimedes was so thrilled by this sudden realization that he leapt right out of the bathtub and ran all the way through the streets of Syracuse, shouting, Eureka! I have found it!

Wait, he ran through the whole city straight out of the bath, with absolutely no clothes on.

That is how the famous legend goes. A Roman writer named Vitruvius wrote the story down about two centuries later. So historians aren't totally sure if the wild running without clothes part is true. But the mathematical breakthrough is totally real.

Okay, so what actually happened when he tested the king's crown?

If you put a block of pure gold in water, a certain amount of water is pushed out. When they put the leafy crown of the exact same weight in water, it pushed out significantly more water.

Oh no, because the silver made the crown bulkier. It took up more volume. The goldsmith really did cheat the king. Exactly.

Now, to be perfectly honest, many historians today think just measuring spilled water from a bowl might have been slightly too messy and imprecise to catch a sneaky goldsmith with absolute proof.

Really? So what do historians think he actually did?

Archimedes most likely used a deeper, more brilliant version of this idea, called his Principle of Buoyancy, where he balanced the crown and the pure gold on a scale while they were completely underwater.

Wow, balancing them underwater? That sounds even cooler.

Yes, but whether it is water spilling over a tub or water pushing back upward, the heart of the idea is exactly the same. An object in water forces exactly its own volume of water out of the way.

That changes how I see everything. If you want to know the volume of anything completely lumpy, you don't need a ruler at all. You just need water.

Yes, this simple idea from ancient Syracuse is exactly how scientists today measure the volume of strange, irregular things. They use it to measure crater-covered rocks from space, weirdly shaped fruit, and tiny engine parts.

Or my bumpy plastic dragon toy. If I fill a bucket to the absolute top edge and push my dragon all the way under. Wait, actually, nobody listening should ever test a full-to-the-brim bath inside the house without a grown-up. Because water damage is definitely not a fun math experiment.

That is very true, Finn. Grown-ups are totally required for any massive displacement experiments in the bathroom.

It is so crazy that a simple bath gave us a scientific tool we still use thousands of years later. Do people still use that word, Archimedes shouted?

They absolutely do. Eureka is an ancient Greek word that translates to, I have found it. And today, when a scientist or an inventor suddenly solves a massive problem they have been stuck on for years, they call it a eureka moment.

I love that. Archimedes didn't just give us a way to measure weird twisty shapes. He gave us the best word ever for suddenly figuring something amazing out.

And that is the true beauty of mathematics. It isn't just a list of numbers on a page. It is noticing that the water rising when you sit in the tub is actually the universe perfectly measuring exactly how much space you take up.

That is going to make jumping into a swimming pool feel completely different.

Alright, Finn, it is quiz time.

Let's see what we remember from the story. Okay, my brain is totally ready. Let's do it. First question.

What is the mathematical name for how much room a thing takes up in the universe? Next one. What was the name of the ancient city on the island of Sicily where this story took place? Question 3. What lighter metal was King Hyro worried the goldsmith had secretly mixed into his solid gold crown? Here's number four.

What ancient Greek word did Archimedes shout when he figured the problem out? Which means, I have found it. Last question. What was the name of the Roman writer who wrote this famous legend down about 200 years later? Great job! Let's see how you did. The first answer is the word for how much space something takes up is volume.

For question two, the ancient city was Syracuse. The answer to number three is the king was worried the goldsmith had mixed in silver. For question four, Archimedes shouted, Eureka! And the final answer is the Roman writer who wrote the story down was named Vitruvius.

You know, I don't think I will ever look at a glass of water with ice cubes in it the same way again. The ice is totally displacing the water.

It really is. Next time you drop an ice cube into a full cup, watch how the water perfectly makes room for it.

If you felt a eureka moment today listening to the story of Archimedes, please leave us a five-star review. Just scroll down and tap the stars.

And tap follow wherever you're listening so the next episode comes straight to you. Oh, and we have sister shows too. History, science and philosophy is not boring. Go give those a listen and a follow as well.

And if you want to shout out on the show, or just want to say hi, email us at hello at kidopoly.com. We'd love to hear from you.

Thanks for exploring with us today. See you next time on Maths is not boring. Bye everyone.

What is volume?

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