The #Ganita of Sandwich Slicing: Are Triangles better than Squares?

Question- How do you like your sandwiches sliced: Triangular or Square?

Triangle

Picture credit: (VIDEO) Tricolour Triangle Tea Sandwiches (theinspirationalnook.com)

Square

(Picture credit: Ribbon Sandwiches (Rainbow Sandwiches) - The Flavor Bender)

To answer this from a Ganita perspective, let's apply some high-school Kshetraganita (the original geometry):

Consider a square bread slice of side length '2a' (area 4a²). Slice it into 4 pieces of equal area a² each in two different ways:

1. 4 squares of side length 'a':

area of each square = a², perimeter of each square = 4a = 2 × 2a, i.e, twice the length of the original side.

2. 4 right-angled triangles:

Confirm the area of each triangle (crust-edge as base) = ¹⁄₂ base × height = ¹⁄₂ × 2a × a =  a²

Next, apply the 'square on the diagonal' result from the Sulbasutras to calculate the length of the diagonal on the original square:

= √(4a^2 + 4a^2) = 2√2 a.
The sides of each triangle are half as long as this diagonal.

Alternatively, using the 'circlometry' of Aryabhata (the original trigonometry), we let φ = the corner angles of the 4 right angled-triangles = 45°.

The original jya of Aryabhata calculates length and not a ratio like sine (it is of the form "R sine φ", where R = radius of a circle), we can obtain the length of the side = 2a×sin(φ) = √2a 

 From this, we get the triangle's perimeter

= base + 2 sides 

= 2a (1 + √2) ~ 2.414 times the length of the original side

The perimeter of the triangular slice is about 21% longer than that of the square slice of the same area. What sorcery is this :)

Since the area of the sandwich filling that is visible to the consumer = perimeter × thickness of the filling,  triangular slices may have better curb appeal as they show off the filling 21% better than square slicing, which is itself twice as good as not slicing at all. 

Each square slice has one full crusty corner, and a crust length of 2a. The triangle is tied on crust length but has two 'half-corners'. So if your kid is fussy about corners, triangles may prevent left-overs in the lunch box. If they don't like crust-edges and the fillings are not exciting, then you may need to go back to the square one.