Cake servings and change of tin
How many slices come out of the tin you own, which size suits your guest list, and what to multiply the recipe by when you swap tins: the factor is between areas, not between diameters.
From the size of the tin to the number of slices, and back: from your guest list to the tin you need. Slice size is a convention, so you pick it and it is always written into the result.
Tiers are separate tins stacked on top of each other and cut one by one, as in a tiered cake. Layers of filling do not count: they make the slice taller, not more numerous. If the tiers are different sizes, work one out at a time and add the results up.
How many slices a tin really gives
The sum is surface of the tin ÷ surface of one slice, rounded down: the leftover is not a serving. A 20 cm (8 in) round tin has 314 cm² (π × 20² ÷ 4), so with generous dessert slices of 40 cm² — roughly 2 × 3 in — you get 7, and with small party slices of 25 cm² you get 12. The three sizes offered are stated conventions, not rules: 40 cm² is the slice eaten with a fork at the end of a meal, 25 cm² the one handed round on a tray, and 12.5 cm² (1 × 2 in) is the wedding finger, the size behind those surprisingly large serving charts. If you have a different slice in mind, type its area and everything follows from it.
Changing tin works on areas, not on diameters
This is the classic kitchen mistake: going from an 8 in to a 9 in tin people multiply by 1.13, which feels sensible and is wrong. Surface grows with the square of the diameter, so the right factor is (23 ÷ 20)² = 1.32. From 20 to 24 cm it is 1.44, not 1.2; from 20 to 26 cm it is 1.69; and the other way round, from 24 to 20 cm it is 0.69 rather than 0.83. Scale by 1.2 instead of 1.44 and the cake loses a fifth of its batter: it comes out flat and dry, and the baking time in the recipe no longer fits. The same applies across shapes: a 20 cm square tin (400 cm²) holds more batter than a 22 cm round one (380 cm²), even though the number printed on it is smaller.
When the sum switches to volumes (and why depth matters)
Leave the depth fields empty and the tool assumes the cake stays the same depth in both tins, which makes the ratio of surfaces enough. Fill in both depths and it moves to volumes — (area × depth) of the new one ÷ (area × depth) of the old one — which is what you want when you deliberately aim for a taller or shallower cake, or when the recipe is written for a deep loaf tin. Rule of thumb: fill a tin two thirds at most, because raising agents and whisked eggs make the batter climb; if the sum tells you to fill it higher, change tin rather than hope.
Eggs do not halve
The factor works for flour, sugar and liquids, but 3 eggs × 1.44 is 4.32 eggs, which do not exist. There are two ways out, and the result shows both. The first is to weigh beaten egg, counting about 50 g per shelled medium egg: beat 5 and take 216 g. The second is to round to 4 eggs and bring the whole recipe to the practical factor that follows, 4 ÷ 3 = 1.33, so the proportions stay consistent. What does not work is rounding the eggs and leaving everything else at the full factor: that is the quickest route to an unbalanced batter.
No numbers on baking times, and the other stated limits
A baking time cannot be produced by a formula: it depends on your oven, on the material of the tin (pale aluminium, dark steel, silicone and ceramic behave very differently), on the batter and on whether the fan is running. The one solid rule follows the depth of the cake: same depth, barely any change; deeper cake, drop the oven by about 10 °C (25 °F) and bake longer; shallower cake, shorten it and start testing early. After that it is the skewer and the colour, not the clock. The other limits: tins are measured inside the rim and sloping ones halfway up; the serving count only looks at the surface you cut and ignores how tall the cake is; and heavily structured recipes (yeasted doughs, cheesecakes, tall sponges) do not scale perfectly in proportion.