To calculate the height of a cliff, a surveyor takes the following measurements. The cliff's shadow is 12m, while at the same time a meter stick has a shadow of 2m. Draw a diagram and find the height of the cliff.

6meters

To find the height of the cliff, we can use similar triangles and the concept of ratios. Here's a step-by-step explanation:

1. Start by drawing a diagram of the situation. Draw a vertical line to represent the cliff, and label it as "Cliff". Then draw a horizontal line to represent the ground, and label it as "Ground". Draw another vertical line from the top of the cliff to the ground, and label it as "Shadow of the Cliff". Finally, draw another vertical line from the bottom of the cliff to the ground, and label it as "Shadow of the Meter Stick". Make sure to label the lengths of the shadows - the shadow of the cliff is 12m and the shadow of the meter stick is 2m.

```
Cliff
|
| Shadow of the Cliff (12m)
| |
| |___
| |
| | Shadow of the Meter Stick (2m)
|____________|___________________
Ground
```

2. Notice that we have two similar triangles in the diagram - one formed by the shadow of the cliff and the cliff itself, and the other formed by the shadow of the meter stick and the meter stick itself. Since these triangles are similar, their corresponding sides are proportional.

3. Set up a proportion to relate the heights of the cliff and the meter stick to their respective shadows. We can write the proportion as follows:

cliff height / shadow of cliff = meter stick height / shadow of meter stick

Let h represent the height of the cliff, and plug in the known values:

h / 12 = 1 / 2

Cross-multiply to solve for h:

2h = 12

Divide both sides of the equation by 2:

h = 6

4. The height of the cliff is 6 meters.