# Using Maths to Make Crispier Roast Potatoes

I was intrigued by a Tweet from Edge Hotel School in Essex about making roast potatoes crispier.

Normally people quarter their potatoes before roasting them so that the four pieces are about the same size. What the students suggested was making diagonal cuts at thirty degrees to the long axis of the potato. Their calculations show that this increases the surface area of the slices by 65%. Here's their diagram.

First let's check their sums. A quick rifle around my vegetable basket showed that a typical potato was 12 cm long, 9 cm wide and 8 cm thick. We can assume that the potato is roughly elliptical  in cross-section to make the sums easier because the area of an ellipse is pi times its radius in either axis.

The long axis and short axis of an ellipse are called the major and minor axes, and so an ellipse has two radii: the semi-major axis (labelled "a" below) and semi-minor axis (labelled "b"). For a circle the values of a and b are equal and so we get a single radius r=a=b.

This isn't just about potatoes; if you study planetary orbits, which are elliptical, you will also see these terms used. Orbits which are very squashed are defined as being very "eccentric" where eccentricity is zero for circles and approaches 1 for extremely elongated orbits,

$$e=\sqrt{1-\frac{b^2}{a^2}}$$

Earth's orbit is very nearly circular, it's eccentricity is 0.02, Mercury's a bit higher at 0.2. For really high eccentricity you have to look at comets that drift into the  inner solar system and have extremely elongated orbits. The orbit of Halley's Comet has an eccentricity of 0.97.

Moving from space back to Earth, let's slice the potato traditionally. First we slice along the long dimension. The major axis is 12 cm and the minor axis is 8 cm so the "radii" are 6 cm and 4 cm. We end up with the potato cut in half and the cut surface area of one piece is

$$6 \times 4 \times \pi = 24 \pi$$

The cut surface area of both pieces is twice that. Then we slice again through the waist of the potato to make 4 pieces. This slice has a cross-section with semi-major axis 4.5 cm (half of 9 cm width) and semi-minor axis of 4 cm (half of 8 cm thickness). Each face of this cut has area

$$4 \times 4.5 \times \pi = 18 \pi$$

Now let's try the new method. The long slice is the same as before but now we have two diagonal cuts at 30 degrees to that first cut. Here's a 3d model:

If you slice a 3d ellipsoid the slice surfaces are ellipses. This cut produces two side wedges each of which has two faces of area 13.8 pi. In total that's:

• Traditional: 24 pi + 18 pi = 42 pi
• New Method: 24 pi + 2 x 13.8 pi = 52 pi

The diagonal slices according to these calculations produce cut surfaces which are 24% larger in area than the traditional method. That's well below the 65% increase that's in the tweet from Edge Hotel School, but is still an improvement. The precise benefit depends on the elongation of your potato with longer potatoes having a larger surface area. In astronomical terms we'd say the more eccentric your potato the crisper your tatties.