Q.Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $\sin^{-1}\left(\frac{1}{3}\right)$.

Idea: Fix the total surface area, express the volume in one variable, maximise it, then read off the semi-vertical angle. Single-variable calculus only.

Let the cone have base radius $r$, slant height $l$, height $h$, semi-vertical angle $\theta$ (so $\sin\theta=r/l$).

• Total surface area (fixed): $S=\pi r^2+\pi r l\ \Rightarrow\ l=\dfrac{S}{\pi r}-r.$
• Volume: $V=\tfrac13\pi r^2 h=\tfrac13\pi r^2\sqrt{l^2-r^2}.$

Maximise $V^2$ (easier, and $V>0$):

Now $l^2-r^2=\left(\dfrac{S}{\pi r}-r\right)^2-r^2=\dfrac{S^2}{\pi^2 r^2}-\dfrac{2S}{\pi}$, so

Differentiate w.r.t. $r$ and set to zero: …

What we are asked

Among all cones with the same total surface area $S$ (curved surface plus base), find the one of maximum volume, and show its semi-vertical angle is $\sin^{-1}(1/3)$.

The semi-vertical angle $\theta$ is the angle between the axis and the slant side, so $\sin\theta=\dfrac{r}{l}$, where $r$ is the base radius and $l$ the slant height.

Set up: use the constraint to remove a variable

For a cone, the total surface area is

Since $S$ is fixed, solve for the slant height:

The volume is

using $h=\sqrt{l^2-r^2}$.

Maximise $V^2$ (avoids the square root)

Because $V>0$, maximising $V$ is the same as maximising

Using $(1)$,

Therefore, with $S$ constant,

Differentiate and find the critical radius

Setting this to zero (with $r>0$, $S>0$):

Check it is a maximum: …

Method: Optimizing Volume Under a Fixed Total Surface Area

This method applies when the cone's total surface area (base circle plus the curved surface) is held fixed, and you must maximize its volume — expressed as a semi-vertical angle.

Steps

Step 1: Write the total surface area constraint and solve it for the slant height

Solve for $l$ in terms of $r$ and the constant $S$:

Step 2: Write the volume in terms of $r$ and $l$, then remove the square root by working with $V^2$

Using $h = \sqrt{l^2 - r^2}$,

Substitute the expression for $l$ from Step 1 so $V^2$ becomes a function of $r$ alone (with $S$ treated as a constant).

Step 3: Differentiate $V^2$ with respect to $r$ and set it to zero …

Common Mistakes

Mistake 1: Using only the curved surface area instead of the total surface area

The question specifies the cone's (total) surface area is fixed, which includes the base: $S = \pi r^2 + \pi rl$. Dropping the $\pi r^2$ term and writing $S = \pi r l$ (as in a curved-surface-only problem) leads to a different, wrong angle — always check whether "surface area" in the problem means curved-only or total.

Mistake 2: Differentiating $V$ with the square root left in, instead of switching to $V^2$

As with any expression containing $\sqrt{l^2-r^2}$, differentiating $V$ directly is far more error-prone than squaring first. Skipping the $V^2$ substitution is a common source of sign and algebra mistakes here. …