# Volume of Pyramid.

Most of the informal proofs about volume of pyramid I have seen involves cutting cube into 3 like this: Then it skips to statement that volume of any pyramid is $$\frac{bh}{3}$$ where $$b$$ is base and $$h$$ is height.

How does cutting a cube into 3 pieces proves that it is true for any pyramid(let's say star pyramid, rectangular pyramid,...) is $$\frac{bh}{3}$$?

• I highly recommend you check out Cavalieri's principle en.wikipedia.org/wiki/Cavalieri%27s_principle Jan 23 at 5:04
• An informal proof: P is the center of the cube. The volume of a cube of edge a is $a^3$. We can see 6 equivalent pyramids, each with a vertex at P and having one of the cube faces as a base. $V_{1pyr}=α.(a^2).(\frac{a}{2})$ $a^3=6.α(a^2)(\frac{a}{2})$. So $α=\frac{1}{3}$ Jan 26 at 21:27

Suppose you know that any pyramid with a square base has volume $$\frac13 bh$$.

Next, suppose the base (with area $$b$$) is a shape made up of $$k$$ squares (with areas $$b_1, \dots, b_k$$). We can chop up the pyramid into $$k$$ pyramids with square bases, which have volumes $$\frac13b_1h, \dots, \frac13b_kh$$. The total volume will be $$\frac13(b_1 + \dots + b_k)h$$, or $$\frac13bh$$.

Next, suppose the base is any other shape. (The pyramid could be a cone, a pentagonal pyramid, whatever.) We can approximate the base arbitrarily well with shapes made up of many tiny squares (that's how your computer screen works). Since all those approximations have volume $$\frac13bh$$, so does the real pyramid.

• I saw that Jacob Claassen's answer already addressed the issue of height, so I did not. Jan 23 at 6:20

Consider what happens when you elongate any 3d shape along one axis by a factor of $$n$$: the volume is multiplied by $$n$$. Now consider what happens when you move the tip of the pyramid so that $$h$$ remains constant: all cross sections of the pyramid remain the same, so the area is the same. It follows that if one pyramid's area is $$bh/3$$, then all similar pyramids are $$bh/3$$.

There are full proofs out there, but I hope this gives a better intuitive understanding of why this works.

Just adding some visual aids for "when you elongate any $$3$$d shape along one axis by a factor of $$u$$"(This is not an answer). Suppose you scale the unit pyramid by $$u=7$$ in $$z$$ axis. It will look like this. Let's change $$z$$ axis to $$u$$ axis where every unit length on vertical axis is $$u$$ times the unit of $$z$$. So in this axis our pyramid will have volume $$\frac{1*1*1}{3}$$ and changing back to original space we get $$\frac{u*1*1}{3}$$.