@mirthfulostrich Hi, thanks for asking! Maybe let's try thinking backwards, and pretend that we are the little ball with diameter of length 5. From his perspective, he wants to get bigger...

M0W2D6-ball-diameter-5-to-10-forum.png

He would like to grow to be like a ball with diameter of $$10,$$ which means he wants to double in all three dimensions:

M0W2D6-ball-diameter-5-to-10-forum2.png

From looking at this, his volume definitely doesn't double.... that would give a snowman. We don't want a snowman!

M0W2D6-ball-diameter-5-to-10-forum3.png

Another illustration of why his volume doesn't double. We don't want a caterpillar!

M0W2D6-ball-diameter-5-to-10-forum5.png

We want something like a ball, with twice the width, twice the length, and twice the height. In terms of balls, it would be like this:

M0W2D6-ball-diameter-5-to-10-forum6.png

The stack of balls has $$2 \times 2 \times 2 = 8$$ balls in it.

Similarly, a large ball with twice the dimensions will have $$2 \times 2 \times 2 = 8$$ times the volume of the ball with diameter $$5.$$

M0W2D6-ball-diameter-5-to-10-forum7.png

$$\text{volume of large ball} = 2 \times 2 \times 2 \times \text{ volume of the small ball}$$

Inverting this to solve for the volume of the small ball, we get

$$\text{volume of small ball} = \frac{1}{2 \times 2 \times 2} \times \text{ volume of the large ball}$$

$$\text{volume of small ball} = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \text{ volume of the large ball}$$
$$\boxed{\text{volume of small ball} = 0.5 \times 0.5 \times 0.5 \times \text{ volume of the large ball}}$$

🙂