# Accurate Volumes with Measuring Spoons

## 1. The Problem

You’re in the middle of baking your favorite chocolate chip cookies. Everything is going swimmingly, until you start to add the salt. The recipe calls for 1/2 of a teaspoon, but you seem to have misplaced your one half measuring spoon. In fact, you can’t find any of your fractional teaspoons. All you have is a hemispherical 1 teaspoon measure, so you resign to estimating the half teaspoon you need. It occurs to you, however, that the task of filling your measuring spoon half-way might be harder than it sounds. You figure you could guess at the half-way mark reasonably well, but filling the spoon to half of its vertical height will not result in filling half of its volume. So the question arises, to what fraction of the height should you fill the teaspoon to end up with half a teaspoon of volume, or any other fraction you want?

## 2. Finding the Volume Formula

To find the volume occupied by the bottom n^{th} of a hemisphere, we can add up the volumes of infinitely thin discs of increasing radius, starting with a disc at height 0, corresponding to the disc of radius 0 at bottom-most point of the spoon, and ending with the disc at the height of .

The volume of each such disc is equal to . To integrate this formula with respect to the distance from the bottom of the spoon (x) we need to find a way to define the radius of each disc (r) in terms of x. To do this, we can assign to each height an angle θ such that a radius θ degrees from the top of the spoon hits the edge at x distance from the bottom of the spoon.

Considering the above diagram and the Pythagorean trig identity we get the following relation for the distance from the bottom of the spoon and the radius of the resulting disc for a spoon of radius R:

or

Now we can rewrite our formula for the volume of the disc as , and now we’re ready to integrate.

## 3. The Integral

## 4. The Function

With the expression for the volume found, let’s try and rearrange it to get a function for the necessary height (h) in terms of the denominator of the desired fraction of volume of the spoon (f). And since we’re working with ratios of the total height and volume, let’s just set R to 1.

So we couldn’t isolate h, but now we have neat little cubic we can plot to get some answers.

## 5. The Results

In our original example with the half teaspoon of salt, the correct height to fill the 1 teaspoon measure is roughly .653 times the entire height of the spoon. Here are the true heights required for some selected fractional volumes:

f | h |
---|---|

1 | 1 |

2 | .653 |

3 | .518 |

4 | .442 |

6 | .355 |

8 | .305 |

These numbers will help us aim, but they don’t let us know the consequences of our measurements falling short. For that, we need to compare the resulting fractional volume of filling up the spoon to a height of only to the desired *volume* of . The equation uses the same integral we derived previously:

By plugging in 3 for f, we see that measuring to only a third of the way up a 1 tbsp spoon would result in a volume .1852 tbsp less than what we bargained for. Baking, though, is much more reliant on the ratios between ingredients, so the percent error might be a better metric of how far off this naive measurement is. For f = 3, the percent error is over 50, meaning the resulting volume is less than half of what it should be, and it only gets worse as f grows. Below is a table and a graph of the rest of the volume errors and percent errors.

f | error | % error |
---|---|---|

1 | 0 | 0 |

2 | .1875 | 37.50 |

3 | .1852 | 55.56 |

4 | .1641 | 65.64 |

6 | .1273 | 76.38 |

8 | .1025 | 82.00 |

## 6. Conclusion

Sadly, there is no neat trick for coming up with the exact correct height to fill a measuring spoon based on the desired volume. For a practical, close-enough rule of thumb, though, if you’re trying for one half, fill your spoon up to two-thirds (which is close enough to the true value of .653) its height . For one third (~ .518) , or one fourth (~ .442) , aim for about half-way up. In general, try to overestimate, keeping in mind that most of a hemisphere’s volume is at the top. Alternatively, invest in some measuring spoons with flat bottoms.