A meme has been going around Facebook asking, "Did you know that it would take an area of just 254 kilometres squared filled with solar panels to power the entire world?" No sources, no calculations from known data, nothing. Just a bald faced assertion.
Really? Let's see if someone with a degree in philosophy, not in engineering, can do a quick back-of-the-envelope calculation.
Wikipedia says,
"In 2008, the world total of electricity production was 20,279 terawatt-hours (TWh). This number corresponds to an average rate of around 2.3 terawatts continuously during the year. The total energy needed to produce this power is roughly a factor 2 to 3 higher because a power plants' efficiency of generating electricity is roughly 30–50%. The generated power is thus in the order of 5 TW. This is approximately a third of the total energy consumption of 15 TW (see world energy consumption)."
https://en.wikipedia.org/wiki/Electric_energy_consumption#Overview
So we need to figure out how much land we would have to cover with solar collectors to collect 5 TW. A typical square meter of the Earth's surface gets 150-300 watts of energy from the sun. https://en.wikipedia.org/wiki/Solar_energy#Potential We will be generous and pick the high value of the range. We also will ignore factors that reduce that value further when it comes to actual usable power, and divide 5 TW by 300. That gets 16,666,666,666.666666667 square meters. Divide that by a million to get square kilometers: 16,666.666666667.
But solar power is not what the electrical generation industry calls "dispatchable": it is not available 24 hours a day regardless of weather, or time of day. The sun doesn't shine 24 hours a day. Let's again be generous and merely double that. We are now at 33,333 square kilometers.
A really efficient solar system, such as Concentrated Solar Power with a Stirling engine (CSP-Stirling) is 30% efficient. https://en.wikipedia.org/wiki/Solar_thermal_energy#Dish_designs Again, we could be generous and assume 1) an all CSP-Stirling installation, and 33% efficiency. But the original requirement was solar panels. Current photo-voltaic panels are somewhere around half that efficiency. So we need to multiply by six to cover the inefficiency. We are now at 200,000 square kilometers.
Because solar power is not dispatchable, it requires backup. The simplest plan is to have a gas fired plant available for when an installation has cloud cover. That would mean that not all the world's power would be generated by solar, which is outside the criteria set by the original post. To meet the criteria, we will add 20% again the area to provide backup, for 240,000 square kilometers. We are now up to an area twice the size of Pennsylvania. https://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_area
Solar is at its most efficient in desert areas, and most cities are on coasts or in lush areas. So you need transmission lines, which take up land area. It is not clear whether the original post included that figure. We will leave that requirement out entirely. But transmission lines have their own energy loses. So let's beef up the generation system to cover those losses. We are now up to 480,000 square kilometers, or four Pennsylvanias.
While we are on transmission lines, we mentioned above that we will assume that solar power generates power half the day. That means you need transmission lines to get power from the day side of the earth to the night side, and to get power from sunny parts of the world to less sunny areas. Again, we omit that calculation.
Also, the electrical generation system must be reliable. As much as possible, the power must be there when the customer turns the switch on. Therefore all calculations must assume the worst case. So we have to assume a hot summer afternoon's demand for any area of the world, not the three in the morning demand. We have ignored that requirement.
We will also ignore growth in demand from developing countries. For example, India's Prime Minister, Narendra Modi, following China's example, expects to expand India's GDP by 8% a year for the next generation. That will require not only more electricity generation capacity, but more reliable capacity than India has now. http://www.economist.com/news/asia/21672359-prime-minister-wants-india-grow-fast-over-next-20-years-china-has-over-past-20
The original post claimed that powering the world with solar power would require "254 kilometres squared" In other words, a square 254 km on a side, or 64,516 square kilometers, roughly the size of West Virginia. The original post is one 7th of a ridiculously over-generous quick back-of-the-envelope calculation.
Note: Since I wrote the above, I have come across several studies that suggest that I underestimate by at least an order of magnitude. http://phe.rockefeller.edu/docs/HeresiesFinal.pdf The more realistic figures is more like 4,800,000 square kilometers, 40 Pennsylvanias, or more than the entire territory of the United states.