Pythagorean Theorem in 3d and 4d

Question

Given right tetrahedron A(0,0,0), B(x,0,0), C(0,y,0), D(0,0,z), what is area of triangle BCD? note: BCD is the only 2d side that is not a right triangle. A tetrahedron has four triangular sides.

Given right pentachoron (4d object) A(0,0,0,0), B(x,0,0,0), C(0,y,0,0), D(0,0,z,0), E(0,0,0,w), what is volume of tetrahedron BCDE?. BCDE is the only 3d side that is not a right tetrahedron. A pentachoron has five tetrahedron sides.

Answer 1

Answer 2

The hypotenuse triangle BCD of right tetrahedron ABCD as described

has areaBCD=√[(xy/2)²+(xz/2)²+(yz/2)²]

The hypotenuse tetrahedron BCDE of right pentachoron ABCDE as described

has volumeBCDE=√[(xyz/6)²+(xyw/6)²+(xzw/6)²+(yzw/6)²]

that is, Pythagorean theorem holds not only for lengths of right triangles

but also for areas of right tetrahedrons and for volumes of right pentachorons.

Answer 3

I can help you with the first question.

The sides of the triangular face are the hypotenuses of the other 3 faces.

Use the theorem of Pythagoras to calculate the length of each of those hypotenuses.

Then use Heron's formula to calculate the area of the face.

Since the 2nd question concerns a 4d object, I expect that there is a definition of area in that 4d space.

Since I do not know how that area is to be computed, I, unfortunately cannot help you with this question.