3D Pythagoras

• 3D Pythagoras refers to the application of Pythagoras’ theorem in three-dimensional space. It’s a natural extension of the Pythagoras’ theorem you’ve studied in 2D, involving right-angled triangles in three dimensions.

• The standard formulae for Pythagoras’ theorem in 2D is a² = b² + c². In 3D, it can be expressed as d² = a² + b² + c² where ‘d’ is the length of the line segment across three-dimensional space and ‘a’, ‘b’ and ‘c’ are the lengths spanning the three dimensions (x, y, z axes respectively).

• This can be particularly useful when trying to determine the shortest distance between two points in three-dimensional space. The first step is to find the distances in the x, y and z directions, then substitute these values into the formula.

• Be careful when using the theorem in 3D as each dimension must be at right angles to the others. Always make sure the dimensions are orthogonal.

• Another important aspect to remember involves the units of measurement. Ensure that all the lengths are in the same measurement unit before inserting them into the formula.

• You can combine the applications of 2D Pythagoras’ theorem and 3D Pythagoras’ theorem. For example, if you need to find the length of a line segment cutting through a rectangular box, you can use 2D Pythagoras to find the length of the line segment across the base, then use that as one dimension in a 3D Pythagoras calculation.

• Also, consider your right angles and be alert to situations where you may need to use trigonometry (sine, cosine, tangent) in combination with Pythagoras, especially when dealing with non-right angles in 3D.

• 3D Pythagoras can often save time in geometric calculations in Physics and Engineering problems, making it a useful tool for those going on to study these subject areas further.

• Practice is key to mastering 3D Pythagoras. The more problems you solve, the more comfortable you will become in applying these principles in different contexts.