What’s the volume of a cube formed by melting three cubes with sides 4, 3 and 2 cm together?

Rooni stood at the entrance with a large metal door; open wide enough to let a person slip in. Holding a small sack by its drawstring, he looked up to read the sign board — ‘Almet Foundry Works’.

He peeked inside and entered the foundry quietly. It was a moderately sized, dimly lit place. The air inside was warm and smelled of hot metal. All sorts of metal castings of different shapes and sizes lay scattered on the floor. At the far end of the factory a man was carefully pouring melted metal into a mould near the furnace.

Rooni made his way around the scattered metal castings almost tripping over them. The man was so engrossed in his work that he failed to notice Rooni approach.

“Mr. Almet?” Rooni asked.

Rooni’s voice startled the man and his hands wavered but he was quick enough to steady the crucible to prevent the hot liquid metal from spilling over.

“Stay back,” said the man.

Rooni stepped back. The man finished pouring the liquid metal into the mould and left it in the holder to cool. He took off the gloves and turned around. “You should always wait for a person to finish his work at hand before interrupting. Now what do you want?” he asked.

“I am really sorry. Are you Mr. Almet?” Rooni asked. The man nodded. Rooni opened the sack he was holding; dug his hand in; took out three cubes and placed them on a table. The red, green and blue cubes gleamed.

Mr. Almet picked up the cubes to examine closely. “Where did you get these from? I haven’t seen such rare metals in ages,” he said.

“I am working on unlocking an ancient artefact and for that I need these cubes melted into one,” said Rooni.

“Eh? An ancient artefact? Whatever you say or do, none of my business. But this job is going to cost you some money,” said Mr. Almet.

“How much?”

“Three pieces of silver.”

“That’s a lot. All I have is a piece each of bronze and copper,” said Rooni.

“Too less. I can’t do the job. Take it somewhere else,” said Mr. Almet.

“But I was told you are the most reasonable person I will ever find,” said Rooni.

“I am being reasonable. I don’t run a charity here,” said Mr. Almet; “These are special metals. They have different and very high melting temperatures. It will require too much fire. And if I am correct these are’t even from this planet.”

“But there surely must be a way. I can pay you later,” said Rooni.

“I don’t do jobs on credit. Who knows whether you will come back to pay me later or not, and I don’t like wasting time in chasing people to collect dues while I could be working in that time. And don’t bother telling me about hiring collection agents. I’ve been down that path before and it isn’t sweet,” said Mr. Aimet.

“I have got to get the cubes melted into one. You have to help me,” said Rooni.

Mr. Almet shifted his gaze from Rooni to the cubes he had been holding all this time. He played with them in his hand and said, “Well, there is one way. I am making an exception just because I find this job challenging and I am very interested in knowing what it will take to melt these rare metals. And I’ll take those bronze and copper pieces too.”

Rooni brightened up. Mr. Almet continued, “You will have to tell me the volume of the cube that will be formed by melting these cubes so that I can make a mould to pour the melted metals in. And I think you know how important this is. If there is any error in your calculations it will be your doing.”

Rooni nodded. He knew that the cube was to be of exact size and weight to fit in the lock to open the artefact. It could neither be small nor big.

“I will need some measuring instruments,” said Rooni.

“You will find everything you need on that shelf,” said Mr. Almet; pointing towards it. He placed the cubes on the table.

Rooni walked over to the shelf. He took a piece of paper, a pencil and measuring scale. Back at the table he sat down on a stool. He lifted the red cube and scratched his head. He looked a bit confused.

Let’s help him.

A cube is a three dimensional geometric object having equal sides. Dices and ice-cubes have all sides equal. If we measure one side of a cube we get the length of each side.

“Hey Rooni, measure one side of each cube and write it down.”

Rooni placed the measuring scale along one side of the Red cube and noted its length. He repeated this for the other two cubes as well.

Red Cube = 4cm

Green Cube = 3cm

Blue Cube = 2cm

Now Rooni has to calculate the quantity of each cube to know how much metal they hold.

How can we know the quantity of a solid?

To know how much quantity a solid has, we can melt it and then measure it in liquid form.

Let’s take an ice-cube of side 2cm and melt it in a measuring cup like the one used to measure liquids. The water from the melted ice-cube will fill the cup up to 8ml.

Or if we take an ice-cube of side 3cm and melt it, the measuring cup will be filled to 27ml.

From this experiment we come to know how the quantity of water is related to the side of an ice-cube.

2 × 2 × 2 = 8

3 × 3 × 3 = 27

From this we can conclude that quantity of a cube is its side × side × side. And this quantity is nothing but its volume. Thus volume of a shape can be described as the amount of space it takes up or the amount of space inside it.

Can we do this without melting it?

Yes we can if we calculate its volume when the length of a side of the cube is known to us.

∴ Volume of a cube = side × side × side = side³

Getting it, Rooni quickly calculates the volume for each cube and writes them down.

Volume of Red cube = 4 × 4 × 4 = 64 cm³

Volume of Green cube =  3 × 3 × 3 = 27 cm³

Volume of Blue cube =  2 × 2 × 2 = 8 cm³

Next, Rooni has to find the space required to hold the three melted cubes.

How do we find the required space to hold these three melted cubes together?

Ice-cubes to the rescue again. If we melt three ice cubes with side 2cm each in a measuring cup, we observe that the level of water stands at 24ml.

Therefore if we melt the three ice-cubes together in a glass, the quantity of water gets added.

8 + 8 + 8 = 24

This means that the total space required to hold the three cubes is sum of their volumes.

To find the volume of a cube formed by melting the three cubes together, all Rooni has to do is add the volumes of the Red, Green and Blue cubes.

Getting the hint Rooni calculates the total volume.

Volume of the final cube = 64 + 27 + 8 = 99 cm³

“I have the volume Mr. Almet. The volume of the cubic mould has to be 99 cubic centimetre,” said Rooni. He got up from the stool and took the paper to show it to Mr. Almet.

Mr. Almet went through the calculations. “You did good. Leave your cubes with me. It will take a day to finish the job. You can come back to collect the final cube the day after tomorrow.”

“Thanks Mr. Almet.” Rooni turned around to leave.

“Wait young man. You forgot to ask for a receipt for the cubes and the job. Let me write one for you,” said Mr. Almet. He took out his job book from the desk drawer and picked up a pen.

“What’s your name?” he asked.


Mr. Almet paused and raised a brow. He repeated the name softly; wrote a receipt; tore out the original from the book and gave it to Rooni. “See you in two days’ time Mr. Rooni.” He smiled.

Rooni pocketed the receipt; thanked Mr. Almet again and left the foundry.

Algorithm to find the volume of a cube formed by melting three cubes together

  1. Input side of first cube (a)
  2. Input side of second cube (b)
  3. Input side of third cube (c)
  4. Volume of first cube (v1) = a × a × a
  5. Volume of second cube (v2) = b × b × b
  6. Volume of first cube (v3) = c × c × c
  7. Volume of final cube (V)= v1 + v2 + v3
  8. Output V

Do you know:

  • 1 millilitre = 1 cubic centimetre.
  • 1 litre = 1000 cubic centimetres = volume of a cube with side 10cm.
By Amar Preet
May 2020