Measuring distances in Minecraft can be quite tedious, but with a few simple guidelines you will never make a mistake again!
Official Units of Measure
Distances in Minecraft are quite easy to measure. Officially , Minecraft uses the metric system, and each block is considered to be 1 cubic meter. To measure long distances, then, simply place a torch or other marker object every ten blocks. Place a second marker at each tenth marker to mark 100 m distances, and at ten of these, you have one kilometer. Try placing a sign at each km, just like road signs in the real world.
Using the Imperial System
Some may wish to measure distances in yards and miles instead of meters and kilometers. Well, it may not be official, but nobody's stopping you, and it's a simulated world anyway.
So, you can simply consider a block as 1×1×1 yard instead. One mile is exactly 1760 yards. Place your initial markers every 16 blocks, and your double markers at every 11 markers rather than every 10. Thus, a double marker measures 16 × 11 = 176 yards, or 1⁄10 of a mile.
One Mile converts to 1609.344m. To simplify, consider a Mile to be 1610 blocks long. So for every 10 blocks, place a marker. Do this 161 times and you will have about a Mile in total. In Real Life, 1 Mile = 1.6 km & 0.625 Mile = 1 km.
If you press F3, the debug screen will give you your present location in X-Y-Z co-ordinates. Measuring distances between two locations or waypoints is as easy as subtraction, if you walk in a cardinal direction. Otherwise you will need to make use of good old algebra to solve the distance. This is not strictly in-game, but it makes a drastic difference in gameplay, avoiding a lot of frustrated wandering. Note that the X and Z coordinates are horizontal and can be positive or negative (the spawn point will be fairly close to 0, 0), but the Y coordinate represents your altitude, and Y=0 is the bedrock floor of the gameworld.
If the measurement is being taken above ground, and lighting the entire path is not necessary, place the markers as above. When 100 m is reached, the 10 m markers can be removed and reused for the next 100 m run. This allows for the path to be constructed without having to count 100 blocks at a time, while still allowing the markers on the completed path to be easily followed without using too much material.
Volume and Surface Area
The formula for the volume of a cube is s3, where s stands for the measurement of one of the cube's side. Since each side of a normal Minecraft block is 1 meter, this would equal 13, which would result in 1m3. (This works the same for yards, or any other unit of length. So do the rest of these comments.)
The formula for the surface area of a cube is 6s2, where s stands for the measurement of one of the cube's side. Since each side of a normal individual Minecraft block is 1 meter, this would equal 6×12, which would equal 6×1, which would result in 6m2.
As you make something, e.g. a house, bigger in all directions, its surface area increases faster than its length, but not as fast as its volume. The surface area tells you how many blocks you'll need for the outer walls, but your interior furnishings will probably increase according to the volume. Of course, shape matters: An 8×8×1 layer of dirt corresponds to a stack of 64 dirt blocks, but so does a 4×4×4 cube, or a 2×2×16 trench or shaft.
Using the Pythagorean Theorem
Sometimes the need arises in which you need to measure distances diagonally. Although it seems complicated, you can use the Pythagorean Theorem to measure this diagonal distance. This geometric formula is normally used to find the length of the hypotenuse in a right triangle. Since Minecraft is basically a world of cubes and right angles, this formula does indeed prove useful. Just picture an imaginary right triangle that covers the distance in your world that you wish to measure (NOTE: The two tips of the triangle that are not right angles should be the points on either end of the diagonal distance that you're trying to measure).
The formula is written as a2+b2=c2, where a and b are the "length and width" of the imaginary triangle and c is the hypotenuse of the triangle, or the diagonal distance that you're trying to measure. The "length and width" variables (a and b, respectively) are actually the X and Z axis of your world (only when measuring diagonal distances on a completely flat surface; any diagonal upwards measuring includes the Y axis). By squaring the X and Z distances, and then by adding them together, you get a number that equals c2, which is actually the square of the distance that you're trying to measure. By finding the square root of this number, you have the diagonal distance (in meters) between two points in your Minecraft world that lie at equal heights on the Y axis.
If, for some reason, one of those two points was higher or lower than the other, your measurement would be off (especially if they were a large vertical distance apart). To fix this, use your previous diagonal distance as a and the vertical distance as b. Using these new numbers, solve for c. That would be your total straight-line distance between the two points.
Here's an example:
Point H (Home) has XYZ coordinates of -35, 68, and 97. Point M (Mineshaft) has XYZ coordinates of 76, 43, and -5. The two points are H(-35,68,97) and M(76,43,-5). Find the total distance between the two places.
That was just the flat part. Now we figure out the vertical component.
Through the Pythagorean Theorem, we have discovered that points H and M are 152.8070679 meters apart.
Alternatively, if you want to include vertical distance immediately (without the hassle of doing the Pythagorean Theorem twice), use a similar formula: a2+b2+c2=d2. In this formula, use a, b, and c for the differences in the X, Y, and Z coordinates (respectively), and d should give the total straight-line distance between the points (which is the same as doing the Pythagorean Theorem twice).