3D Pythagoras

Understanding 3D Pythagoras

• The 3D Pythagoras theorem extends the concept of the 2D Pythagorean theorem into three dimensions. It’s used to calculate the distance between two points in 3D space.
• It involves the use of the Pythagoras theorem on two right-angled triangles consecutively.

Calculation

• Firstly, apply the 2D Pythagorean theorem to one of the right-angled triangles to get the hypotenuse. That is the square root of the sum of the squares of the two shorter sides.
• Secondly, take this result (the hypotenuse of the first 2D triangle), square it and add it to the square of the third side (in the 2nd right-angled triangle) to get the square of the final distance.
• The result is then square rooted to find the actual distance.

Application

• This is extremely useful if you want to know the shortest distance between two points in 3D, such as on a cube, rectangular prism, cuboid, pyramid or cone.
• It’s also used extensively in maths-related fields such as physics, where it’s essential to calculate the distance between objects in 3D space.

Example Problem

• If you have a rectangular prism of dimensions 4cm, 3cm and 6cm, the longest diagonal, that is the distance from one corner to the furthest corner, can be calculated using the 3D Pythagorean theorem.
• The first step would involve calculating the hypotenuse of the base which is sqrt(4^2 + 3^2) = 5cm.
• Then, take this result, and use the 3D Pythagoras formula again to find the longest diagonal. So the longest length is sqrt(5^2 + 6^2)= sqrt(61)cm.

Keep in mind the 3D Pythagorean theorem is not meant for use with shapes that don’t include right angles like spheres or complex, irregular 3D shapes. Repeat the application of this theorem until it is fully understood and is second nature. Seen in the practical sense, it turns geometry into a much simpler subject.