Estimate this
Kamal Lodaya, The Institute of Mathematical Sciences, Chennai

Question: If all the humans in the world were crammed together, how much area would we occupy?
Ans: There are 600 crore of us humans. How many can we cram into a square metre (that is, a square which is 1 metre by 1 metre, or about 3 feet by 3 feet)? It is difficult to say, but certainly three of us can be comfortably placed inside such a square. Putting ten of us inside a square of that size would be very difficult. So let us {estimate} that the number of people which can be crammed into a square metre is six.
This means that 600 crore of us can fit into 100 crore square metres. That sounds like a lot, let us see how big it is.
A hundred crores is 10^9, 10 multiplied by itself 9 times. 1 square kilometre is 1 kilometre x 1 kilometre = 10^3 metres x 10^3 metres = 10^6 square metres. So 100 crore square metres is 10^9/10^6 = 10^3, or a thousand, square kilometres.
A thousand square kilometres sounds like a lot, let us see how big a square that would need.
If we take a square which has each side 30 kilometres long, its area would be 30 x 30 = 900 square kilometres. So a square which is a little over 30 kilometres long should be enough.
A big city like Chennai or Chandigarh is more than 30 kilometres long and 30 kilometres wide. A mega-city like Delhi or Mumbai is even bigger. So the answer is: all the humans in the world could be fitted into one city.
You might be thinking that the question is ridiculous. We cannot just cram people, they need some space to live in... So here is a question to think about.

Question: Supposing we gave every family a house to live in. Now how much area do we require to house every family in the world?

Answer: For the answer, we first need to estimate the size of an average family. In India it is close to 4, but in the western world, it is closer to 3. So let us say an average family has 3 people but we will compensate by giving each family not only a house but a yard around the house. There are 600 crore people in the world so there are 200 crore = 2 x 10^9 families.
How much area should a family get? A house would be around 10 metres long, a football field would be around 100 metres long. We will take 30 metres, so that there is a generous yard all around the house. So the area of a house is 30 x 30, or about 1000 square metres. All the families together give 2 x 10^{12} square metres.
A square kilometre is 1000 x 1000 = 10^6 square metres, so that is 2 x 10^6 square kilometres. That sounds a lot but it is the area of a triangle which is 2000 kilometres high and has a base of 2000 kilometres.
The southern half of India (the Deccan region) has the shape of a triangle of about this size. So all the world's families can be comfortably housed in south India.
Question: Here is a more frivolous question for you to think about: How may tennis balls would it take to encircle the Earth?
Answer: We need to estimate the size of a tennis ball and the size of the Earth. Well, from what I remember, a tennis ball is around 5 centimetres in diameter. For the Earth we could look up a geography book or find it on the internet, but the whole idea of this article is to estimate numbers, not look them up.
But we will need some geographical data. Let us assume you know, as I do, that Chennai is on latitude 13 degrees north and Delhi is around latitude 33 degrees north or 20 degrees further north. I also know that the Tamil Nadu Express takes 33 hours to cover the distance from Chennai to Delhi, and given the average speed of trains in Indian railways, that would mean that the distance is around 2000 kilometres.
So 20 degrees of latitude is 2000 kilometres, or 10 degrees of latitude would be 1000 kilometeres. This means that 360 degrees of latitude would be 36000 kilometres. That is the circumference of the Earth. In centimetres it is 36 x 10^3 x 10^5 = 36 x 10^8. Dividing by 5 gives us 7.2 x 10^8. So around 72 crore tennis balls, put side by side, should be sufficient to encircle the Earth.
Here is a question for ardent tennis fans: How many people play tennis? How many tennis balls will there be in the whole world?

Based on
Guesstimation
by
Laurence Weinstein and John A. Adam