# How deep is the liquid in a half-full hemisphere?

I have a baking recipe that calls for 1/2 tsp of vanilla extract, but I only have a 1 tsp measuring spoon available, since the dishwasher is running. The measuring spoon is very nearly a perfect hemisphere.

My question is, to what depth (as a percentage of hemisphere radius) must I fill my teaspoon with vanilla such that it contains precisely 1/2 tsp of vanilla? Due to the shape, I obviously have to fill it more than halfway, but how much more?

(I nearly posted this in the Cooking forum, but I have a feeling the answer will involve more math knowledge than baking knowledge.)

• after I answered, I found here that the answer is to fill it by the fraction $1-2\cos(\frac49\pi)$ Sep 22, 2019 at 21:51
• Not really relevant to the mathematics, but: just eyeball it. Half a teaspoon of vanilla extract, one way or another, ain't gonna make that much difference. In fact, just put in a full teaspoon. Then, do yourself a favor and add some mace and clove, too. ;) Sep 23, 2019 at 13:55
• @RandomAspirant Do you need to comment that on every answer as well? Sep 23, 2019 at 14:38
• Just use a second spoon - fill the first one completely, then pour from it into the second until they're even... Sep 23, 2019 at 16:55
• Do we allow housework problems? Sep 23, 2019 at 21:53

Assuming the spoon is a hemisphere with radius $$R$$,

let $$x$$ be the height from the bottom of the spoon, and let $$h$$ range from $$0$$ to $$x$$.

The radius $$r$$ of the circle at height $$h$$ satisfies $$r^2=R^2-(R-h)^2=2hR-h^2$$.

The volume of liquid in the spoon when it is filled to height $$x$$ is $$\int_0^x\pi r^2 dh=\int_0^x\pi(2hR-h^2)dh=\pi Rh^2-\frac13\pi h^3\mid_0^x=\pi Rx^2-\frac13\pi x^3.$$

(As a check, when the spoon is full, $$x=R$$ and the volume is $$\frac23\pi R^3,$$ that of a hemisphere.)

The spoon is half full when $$\pi Rx^2-\frac13\pi x^3=\frac13\pi R^3;$$ i.e., $$3Rx^2-x^3=R^3;$$

i.e., $$a^3-3a^2+1=0$$, where $$a=x/R$$.

The only physically meaningful solution of this cubic equation is $$a\approx 65\%.$$

• What the hell, call it $\frac23$. Everybody likes vanilla. Sep 22, 2019 at 21:46
• And they say calculus is of no use in real life... Sep 23, 2019 at 13:41
• @DoktorJ but that principle leads to a divergent sequence! Sep 24, 2019 at 19:19
• @JosephSlote ah but it'll be a very delicious divergent sequence! Sep 24, 2019 at 20:15
• @HenryvanMegen you must not have tried Holly's cookies then! Now with 0.73% more vanilla extract, thanks to calculus. Sep 25, 2019 at 13:42

It may be surprising that the problem actually admits an analytic solution.

A spherical cap is the difference between two overlapping cones, one with a spherical bottom and the other with a flat bottom, i.e.

$$V = \frac{2\pi}{3}r^2h - \frac{\pi}{3}(2rh-h^2)(r-h) =\frac{\pi}{3}(3rh^2-h^3)$$

which, with half of the semisphere volume $$V=\frac{2\pi}{3}r^3$$, becomes

$$\left(\frac rh \right)^3 - 3\frac rh+1=0$$

Let $$\frac rh = 2\cos x$$ and compare with $$4\cos^3 x -3\cos x -\cos 3x=0$$ to obtain $$x=40^\circ$$. Thus, the depth $$h$$ as a fraction of the radius $$r$$ is

$$\frac hr = \frac{1}{2}\sec 40^\circ$$

• That was a big surprise for me! Also a bit surprising was that your $\dfrac{1}{2\cos 40^\circ}$ is equal to J.W.Tanner's $1-2\cos(\frac49\pi)$ (in a comment to the OP). Sep 23, 2019 at 16:16
• @TonyK - I knew of the close-form result, but was also surprised of a different form from his, until convinced myself numerically Sep 23, 2019 at 16:34

It makes things a bit simpler if we turn your measuring spoon upside down, and model it as the set of points $$\{(x,y,z):x^2+y^2+z^2=1, z\ge 0\}$$. The area of a cross-section at height $$z$$ is then $$\pi(1-z^2)$$, so the volume of the spoon between the planes $$z=0$$ and $$z=h$$ is

$$\pi\int_0^h(1-z^2)dz = \pi\left(h-\frac13h^3\right)$$

The volume of the hemisphere is $$\frac23\pi$$, and we want the integral to be equal to half this, i.e. $$\pi\left(h-\frac13h^3\right)=\frac{\pi}{3}$$ or $$h^3-3h+1=0$$ This cubic equation doesn't factorize nicely, so we ask Wolfram Alpha what it thinks. The relevant root is $$h\approx 0.34730$$. Remember that we turned the spoon upside down, so you should fill it to a height of $$1-h=0.65270$$, or $$65.27\%$$.

• "It makes things a bit simpler if we turn your measuring spoon upside down" Then there isn't any liquid in the hemisphere. Sep 23, 2019 at 21:47
• @Acccumulation One could easily turn it upside-down to measure it, then turn it right-side-up when filling it Sep 23, 2019 at 23:03
• What do you mean by "doesn't factorize nicely"? In my view, $$h^3-3h+1=\left(h-2\cos\frac{2\pi}9\right)\left(h-2\sin\frac{\pi}{18}\right)\left(h+2\cos\frac\pi9\right)$$ is quite a nice closed-form factorization. Not in radicals, but why would anyone want them :) Sep 24, 2019 at 6:27
• @Ruslan How did you find that?
– Ovi
Sep 25, 2019 at 0:39
• @Ovi well, I found the first root with Wolfram Mathematica's Solve + FullSimplify, and the second and third by combination of FullSimplify on the additive terms of the solution returned by Solve and then ExpToTrig to get the expressions like $(-1)^{8/9}$ to trigonometric form. But that was a lazy approach. The more general way is to use the algorithm given in this page (page is in Russian, but I guess if you just follow the formulas, you'll get it: the Vieta solution is sufficient here). Sep 25, 2019 at 5:32

Without loss of generality we assume the radius of the sphere to be $$1$$

The volume of the liquid is found by an integral $$V= \int _{-1}^{-1+h} \pi (1-y^2 )dy$$

and you want the volume of the liquid to be half of the hemisphere which is $$\pi/3$$

After evaluating the integral and solving the equation I have found $$h=0.65270365$$ That is a little bit more than half as expected.

Alternative: use two teaspoons.

Use water as you develop your skill. Fill tsp A, and pour into tsp B until the contents appear equal. Each now contains half a tsp. And now you know what half a tsp looks like in practice.

And you don't have to calculate cosines against thumb-sized hardware.

• -1. Holly doesn't have two teaspoons. Sep 26, 2019 at 22:19

Note about eyeballing: Your eye's reference is the surface of the spoon, so when you eyeball you may actually be measuring along the arc from the bottom of the spoon to its top edge.

That is, your eye may be watching the red curve, not the blue line:

Using the 65.27% from other answers, the depth measured along the red curve is $$\frac{\arccos(1 - 0.6527)} {90\deg }\approx 77.42\%$$

So to the eye, the "depth" of a half-full spoon may look like more like three quarters than two thirds.