You gain nothing if create a hill on your land
The Sunday Nation
Nairobi,
09 September 2012
Kimari wa Kagiri wants to know if he can increase his acreage by creating hills in his shamba. “How much soil do I need to increase it from 10 to 15 acres?”
My surveyor tells me that land measurements are made using horizontal distances; not along the slope. The logic is that this is the effective usable space. Thus an acre on a hill looks bigger than that on a flat plane.
Suppose you want to put up a house on a plot that is on the sloping land, would you use the slanted ground as your floor…and build slanting walls? Of course not! The first step would be to level the surface horizontally.
The same argument applies when farming; the recommended separations between plants are measured horizontally. The reason being that the plants will grow vertically upwards (and downwards underground). Thus Kagiri’s proposition would be a futile exercise.
Nevertheless, it would be interesting to know how much “extra” land one gets when the land is located on slanting terrain. For simplicity, suppose the profile is a uniform incline and that the plot is rectangular in shape with the one side lying along the slanted direction.
All hills are not equal: some are steeper than others. But for the purpose of illustration, suppose the slope is a “one-in-five”. That is, for every five metres (or feet, or whatever) covered horizontally (note; NOT along the slanting ground), you climb up vertically by one metre (or foot or whatever, respectively).
Now imagine a vertical cross-section cut on this land along the sloping direction. It will reveal a right angled triangle with a horizontal side, a vertical and a slanted one. Then immediately, we recognise that we can make use of Pythagoras’ Theorem – the sum of the squares of the horizontal and vertical sides is equal to the square of the slanting one.
For our “one-in-five” slope: one squared is one; five squared is 25; the sum of these two is 26; and the square-root of 26 is 5.099. Therefore, for every one metre (or foot or whatever) of vertical rise, the distance covered along the sloping surface is about 5.1 metres (or feet or whatever).
The ratio of the sloping the distance to the horizontal one is 1.02 (that is, 5.1 divided by 5). Therefore, the land on this terrain will be about 2 per cent more than an equivalent area measured on a flat level ground.
So with this knowledge; what kind of hill would Kimari need to build in order to gain 50 per cent extra area on his 10 acres? The horizontal distance is 10 “whatever” and he needs to make a sloping length of 15 “whatever”.
Bearing in mind that we now have the desired length along the slope, we apply Pythagoras’ theorem thus: 15 squared is 225; 10 squared is 100; the difference is 125; the square-root of 125 is 11.2. Therefore the gradient of the hill must be an “11.2-in-10”. Now that’s a fairly steep hill and let’s not forget that he will not gain anything by creating it!
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