Scale map height

Question

I am trying to work out a 3D model map that I'd like to make with a group of primary/middle school students. I am having difficulty scaling the height in the map proportionally based on the limitations of our reproduction size. I have been trying to read up on the appropriate trigonometry but can't seem to figure it out for my purposes. I'm sorry for the very simplistic question but am very hopeful that this group of experts can help me bring this project to reality for my students.

We want to make a reproduction scale model of New Zealand using a board $200$cm long by $70$cm wide. I have calculated our current scale for this as $1:800000$. So for all the 2d distances on the model I am using $1km = 1.25mm$ on the map. The problem I'm having is height calculation. Mount Aoraki/Cook is our highest mountain at $3.724km$ high. Given that the actual length of NZ is $1600km$ and the widest part of the islands is $400km$, how would I calculate a proportional height based on the scale above?

Answer 1

If you could approximate Aoraki with height $h$ as a square region and call the length of one of its sides $s$, then you could take a ratio of its length to its height since length, height, and the $2$-dim scaling are known

$$\frac{s}{1/800\space000}=\frac{h}{h_{scale}}\implies h_{scale}=\frac{1}{800\space000}\frac{h}{s}$$

If $h < s$ then its height out of the map is less than its approximated side length

If $h=s$ then its height out of the map is the same as its side length

etc.

The region of the mountain seems to align with the direction of New Zealand, so directional issues with scaling shouldn't be too much of a factor

Answer 2

The problem is that relative to its size, the earth’s surface is basically smooth. At a scale of $1\text{ km} = 1.25\text{ mm}$, Mt. Aoraki is all of $4.66\text{ mm}$ tall. That’s not going to make for a very interesting scale model—it will end up being more or less flat.

I would suggest that you choose a different scale for elevations and then explain to your students what’s going on and why you’re doing it that way. So, pick a convenient height for the peak in your model and scale all of the other elevations proportionally. You shouldn’t need any trigonometry to do this, just simple ratios (pax those who would point out that trigonometric functions are ratios themselves).