TITLE: How deep is the liquid in a half-full hemisphere?
QUESTION [80 upvotes]: 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.)

REPLY [3 votes]: 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.