Deriving Formulae for Cone, Pyramid and Sphere

[DB]

I'm just a little curious with these formulas because I like to understand a formula before I use it. There's no doubt I've been using these since gr3 but I would like a mathmatical derivation of them.

$$V_{cone}=\frac{A_bh}{3}$$
$$V_{pyramid}=\frac{A_bh}{3}$$
All I would like to know here is why by 3?

And just
$$V_{sphere}=\frac{4\pi r^3}{3}$$
With this one I have no clue why it is what it is. There are $4\pi$radians in a sphere maybe??

Thanks in advance.

 

[One-D]

hm do you mean where does 3 come from?
in integration (calculus), we can find the formula by integration.
for ball, find antiderivative of p f(x)^2 dx

 

[Data]

Well, a cone of height $$h$$ and with base radius $$\alpha h$$ can defined by $$T = \{ (x,\ y,\ z) \ | \ (\alpha z)^2 \geq x^2 + y^2,\; 0\leq z \leq h \}$$. Clearly it is a solid of revolution (it is symmetric about the z-axis). Its volume is

$$V_{\mbox{cone}} = \int \int \int_T dV = \int_0^h \int_{-\alpha z}^{\alpha z} \int_{-\sqrt{(\alpha z)^2 - x^2}}^{\sqrt{(\alpha z)^2 - x^2}} \ dy \ dx\ dz = 2 \int_0^h \int_{-\alpha z}^{\alpha z} \sqrt{(\alpha z)^2 - x^2} \ dx \ dz = 2\int_0^h {(\alpha z)^2 \pi \over 2} \ dz = \alpha^2 \pi \left[{z^3\over 3}\right]_0^h = {\alpha^2 \pi h^3 \over 3}$$

From here all you need to notice is that the base area is $$A_b = \alpha^2 h^2 \pi \Longrightarrow V_{\mbox{cone}} = {\alpha^2 \pi h^3 \over 3} = {A_b h \over 3}$$ as we wanted. Note that the third-to-last step is valid:

$$\int_{-a}^a \sqrt{a^2-x^2} dx = \int_{-{\pi \over 2}}^{\pi \over 2} a^2 \cos^2{t} \ dt = a^2\int_{-{\pi \over 2}}^{\pi \over 2} {1\over 2}\left[1+\cos{2t}\right] \ dt = {a^2 \over 2}\left[t + {\sin{2t} \over 2}\right]_ {-{\pi \over 2}}^{\pi \over 2} = {a^2 \pi \over 2}$$


This is, incidentally, the area of a semicircle of radius $$a$$, which is to be expected since $$0 \leq y \leq \sqrt{a^2-x^2}, -a \leq x \leq a$$ defines such a semicircle!

The pyramid can be done in a similar manner, or by appealing to its geometry. You can try it :)

The volume of a sphere is also found in the same way, although this one's easier because you can just use spherical coordinates (who would have guessed!). A sphere of radius $$r$$ can be defined in polar coordinates to be just $$S = \{ ( \rho, \ \phi, \ \theta ) \ | 0 \leq \rho \leq r, \ 0 \leq \phi \leq \pi, \ 0 \leq \theta < 2\pi \}$$. Using the Jacobian transformation we find that since our coordinate transformation is $$x = \rho \sin{\phi} \cos{\theta}, \ y = \rho \sin{\phi} \sin{\theta}, z = \rho \cos{\phi}$$ we get $$dV = \rho^2 \sin{\phi} \ d\rho \ d\phi \ d\theta$$ and so our volume is

$$V_{\mbox{sphere}} = \int \int \int_S dV = \int_0^{2\pi} \int_0^\pi \int_0^r \rho^2 \sin{\phi} \ d\rho \ d\phi \ d\theta = 2\pi \int_0^\pi \sin{\phi} \ d\phi \int_0^r \rho^2 \ d\rho = 2\pi\left[-\cos{\theta}\right]_0^\pi\left[{\rho^3\over 3}\right]_0^r = (2\pi)(2){r^3 \over 3} = {4 \pi r^3 \over 3}$$

again as we wanted. As to your question about $$4\pi$$ radians in a sphere: not quite, but very close. There are $$4\pi$$ of what we call steradians in a sphere. These are a "unit" (and I use the term loosely, because neither radians nor steradians are really a unit) for a sort of two-dimensional angular arc.