A frustum is what remains when you slice the top off a cone or pyramid with a cut parallel to the base — like a truncated cone. Water towers, buckets, and hopper bins are frustum shapes. The formula looks unusual at first but notice that $R^2 + Rr + r^2$ averages the three "sizes" of the cross-section — it reduces to the cone formula ($r=0$) and to a cylinder ($R=r$) as special cases. For the lateral (curved) surface area of a conical frustum, you need the slant height $l=\sqrt{h^2+(R-r)^2}$ — the distance measured along the slanted face. A prismatoid (any solid with parallel top and bottom faces) uses Simpson's Rule: $V=\frac{h}{6}(A_{\text{top}}+4A_{\text{mid}}+A_{\text{bot}})$.
A closed cylindrical tank with radius 2 m and height 5 m needs two coats of anti-corrosion paint. One liter of paint covers 8 m². How many liters are needed?
A water reservoir is shaped like a conical frustum with lower radius 4 m, upper radius 7 m, and depth 3 m. The water depth is 2 m from the bottom. Find the volume of water.
At depth $y$ from the bottom, the radius varies linearly: $r(y)=R_{\text{bot}}+\frac{(R_{\text{top}}-R_{\text{bot}})}{H}\cdot y=4+\frac{3}{3}y=4+y$.
At $y=0$: $r=4$ m. At $y=2$ m: $r=6$ m. Use frustum formula with these values and $h=2$.
