Q.Show that the right circular cone of least curved surface and given volume has an altitude equal to $\sqrt{2}$ time the radius of the base.

Idea: Fix the volume, write the curved surface area as a function of one variable, and minimise it with single-variable calculus (no Lagrange needed).

Let the cone have base radius $r$, height $h$, slant height $l=\sqrt{r^2+h^2}$.

• Curved surface area: $S=\pi r l=\pi r\sqrt{r^2+h^2}$.
• Given (fixed) volume: $V=\tfrac13\pi r^2 h\ \Rightarrow\ h=\dfrac{3V}{\pi r^2}$.

Minimising $S$ is the same as minimising $S^2=\pi^2 r^2(r^2+h^2)=\pi^2 r^4+\pi^2 r^2 h^2$. Substitute $h^2=\dfrac{9V^2}{\pi^2 r^4}$:

Differentiate and set to zero:

Now use $h^2=\dfrac{9V^2}{\pi^2 r^4}$, so …

What we are asked

Among all right circular cones of the same volume, find the one with the smallest curved surface area, and show its height $h$ satisfies $h=\sqrt2\,r$, where $r$ is the base radius.

Setting up the two facts

For a right circular cone with radius $r$, height $h$ and slant height $l$:

• slant height: $l=\sqrt{r^2+h^2}$,
• curved surface area (what we minimise): $S=\pi r l=\pi r\sqrt{r^2+h^2}$,
• volume (what is fixed): $V=\dfrac13\pi r^2 h$.

The volume $V$ is a given constant, so the two variables $r$ and $h$ are not free — they are linked. From the volume,

This lets us turn $S$ into a function of the single variable $r$.

A neat trick: minimise $S^2$

Square roots are awkward to differentiate, and $S>0$, so minimising $S$ is exactly the same as minimising $S^2$:

Now put in $(1)$, $\pi^2 r^2 h^2=\pi^2 r^2\cdot\dfrac{9V^2}{\pi^2 r^4}=\dfrac{9V^2}{r^2}$:

Find the critical point

Differentiate with respect to $r$:

Set $f'(r)=0$:

Read off the ratio $h/r$ …

Method: Minimizing a Surface Expression That Contains a Square Root (the "Minimize $S^2$" Trick)

Whenever the quantity you must optimize involves a square root — most often because it comes from a slant height $l=\sqrt{r^2+h^2}$ — differentiating it directly is messy and error-prone. This method removes the square root before differentiating.

Steps

Step 1: Write the objective and the constraint using the solid's standard formulas

For a right circular cone with base radius $r$, height $h$, and slant height $l=\sqrt{r^2+h^2}$:

Step 2: Use the constraint to eliminate one variable

Solve the fixed-volume equation for $h$ in terms of $r$ (and the constant $V$):

Step 3: Minimize $S^2$ instead of $S$

Since $S>0$, the value of $r$ that minimizes $S$ is exactly the value that minimizes $S^2$ — and $S^2 = \pi^2 r^2(r^2+h^2)$ has no square root, so it differentiates cleanly. Substitute the $h$ from Step 2 to get $S^2$ as a function of $r$ alone.

Step 4: Differentiate, set to zero, and solve …

Common Mistakes

Mistake 1: Differentiating $S = \pi r\sqrt{r^2+h^2}$ directly instead of squaring first

Differentiating a product involving a square root (using the product and chain rules together) is far more error-prone than differentiating the algebraically simpler $S^2$. A student who insists on differentiating $S$ directly is much more likely to drop a factor or mishandle the chain rule.

Mistake 2: Confusing "curved surface area" with "total surface area"

This chapter contains near-identical problems that ask for the curved surface ($\pi r l$) versus the total surface ($\pi r^2 + \pi rl$, including the base). Using the wrong formula for "surface area" gives a completely different optimal ratio between $h$ and $r$ — always re-read exactly which surface the question means. …