In the UK, paper sizes have been metric for many years, using the international A series standard. Most office workers are familiar with A4, which is used on laser printers, inkjet printers, photocopiers, and for letterheaded stationery.
The A series of paper is based on the base size of A0 having an area of 1 square metre. All A-sized pages have a ratio of long side to short side equal to the square root of 2, which is approximately 1.414.
This means that when you halve the size of a sheet of A4 (along the longest side), the resulting two smaller pages will have the exact same ratio of long side to short side of 1.414. The new page sizes add 1 to the number as they are halved in size, and if you halve the size, then 1 is added to the size, e.g. half of A4 is A5, and if you double the size then add 1 to the size, e.g. A3 is double the size of A4.
A4 sheets are 297 by 210 mm in size, A3 are double in size at 297 × 420 mm. A5 is 148 × 210 mm.
The diagram below shows the relationship between A4 and A3. Note that an A3 sheet has double the area of an A4 sheet of paper, and that 2 A4 sheets fit exactly into the size of an A3 sheet.
A0 paper size has an area of 1 m² (one square metre), it follows then that A1 size must be half a square metre in area, A2 is one quarter of 1 m², A3 is one eighth, and A4 is one sixteenth.
The weight of paper is expressed in grams per square metre, written properly as g/m², however it sometimes is written as gsm.
If the paper you are using has a weight of 80 g/m², then if you have enough paper to cover an area of one square metre, such as 16 sheets of A4, those sheets will weigh 80 grams.
This is helpful in posting letters, for example, as if the paper has a weight of 80 g/m², then each individual sheet of A4 is 80 grams divided by 16 sheets, which is 5 grams for each sheet of A4 paper.
Standard first class postage in the UK covers weight of the letter/envelope up to 60 grams, so it should be no more than this to use a standard first class stamp. If we say that the envelope weighs 10 grams, then you can put 10 sheets of A4 inside (50 grams). Very handy to know.
The Royal Mail has been metric for a long time, giving prices for set intervals of weights in grams. For example, a first class letter in the UK weighing up to 60 grams, will cost the standard rate for first class post, but if it weighs over 60 grams up to 100 grams, then the cost will be more, and then each 50 g increase will increase the price to the next level. See their guidance notes on size and weight, all of which are in metric.
Read more about paper sizes here. A4 letters can be folded into thirds to fit neatly inside the standard DL-sized envelope, which is 110 × 220 mm.