Meticulously Cutting Fruits

Relevant wiki: Surface Area of a Sphere

According to Archimedes' Hat-Box Theorem , the lateral surface area of the sphere portion will equal to the lateral surface area of the cylinder with the same height as the portion and the same radius as the whole sphere.

Considering the image above, let $\theta$ be the angle the portion's rim makes with the vertical axis. In the image, $\theta$ turns out to be $a$ and $b$ for the 2 right triangles in the sphere's portion. Then the surface area (A) of the sphere's portion can be calculated as the integration of infinitesimal areas of lateral cylindrical layers.

For each layer, the circumference equals $2\pi r(\sin \theta)$ . With an infinitesimal change, $\mathrm{d}\theta$ , the height of each layer is approximately $r(\sin (\mathrm{d}\theta)) \approx r(\mathrm{d}\theta)$ .

Hence, the surface area can be calculated as:

$A = \int_a^b (2\pi r(\sin(\theta)))(r) \ \mathrm{d}\theta = 2\pi r^2 \int_a^b\sin(\theta) \ \mathrm{d}\theta = 2\pi r^2 [\left.-\cos(\theta) \right|_a^b] = (2\pi r)r[\cos (a) - \cos (b)]$

Since the thickness of each slice $h$ is the difference between the vertical sides of both triangles, it can calculated as:

$h = (r\times \cos (a)) - (r\times \cos (b)) = r[\cos (a) - \cos (b)]$

Substituting this term in the previous equation, we will get:

$A = (2\pi r)r[(\cos (a) - \cos (b))] = 2\pi rh$

This is clearly the formula for the cylindrical cucumber slice's lateral surface area with radius $r$ and $h$ . As a result, each tomato slice will have the same lateral surface area as the cucumber slice as long as the slice's thickness is the same!

Hence, each slice of both kinds have the same lateral surface area.