Question people ask: how many calories does my dog need a day?

The short answer: a neutered healthy adult dog needs about 1.6 x 70 x (weight in kg)^0.75 kcal a day: about 630 kcal at 10 kg and 1436 kcal at 30 kg. The popular shortcut, 30 x weight + 70, is close for mid-sized dogs but gives a 45 kg dog 16.8% more, about 326 kcal a day.

What we found

The resting energy requirement (RER) of a dog is 70 x (body weight in kg)^0.75 kcal a day; for a 10 kg dog that is 393.6 kcal, which Ohio State's veterinary nutrition service rounds to about 400.
The shortcut 30 x (weight in kg) + 70 gives 370 kcal for the same 10 kg dog, 6.0% less.
The MSD Veterinary Manual allows the shortcut only for animals over 2 kg and under 45 kg. Inside that range it is within 2.5% from 15 to 20 kg, -6.0% at 5 kg and 10 kg, +8.1% at 30 kg and +16.8% at 45 kg.
A neutered healthy adult dog needs about 1.6 x RER a day (MSD): 630 kcal at 10 kg, 1059 at 20 kg, 1436 at 30 kg and 1946 at 45 kg.
At 45 kg, using the shortcut for a neutered adult gives 2272 kcal against 1946: 326 kcal a day more.
The MSD multipliers for adult dogs are 1.8 x RER intact, 1.6 x RER neutered and 1.4 x RER for an obesity-prone dog; puppies get 3 x RER up to 4 months and 2 x RER after.
A dog 10 times as heavy needs about 5.62 times the resting energy, not 10 times, because the requirement grows with weight to the 0.75 power (10^0.75 = 5.62).

Why the figures differ (explanations you may use; do not add others)

Why the shortcut drifts: it is a straight line, and the requirement is a curve (weight to the 0.75 power); a line can match a curve closely only over a short stretch, which is why MSD limits it to 2 to 45 kg [2].
What RER covers: the energy of a healthy, fed animal at rest in a comfortable temperature, including recovering from activity and feeding [2].
What the multiplier adds: the maintenance requirement covers a moderately active animal, getting and digesting its food and normal activity, and depends on age and neuter status [2].
Why any figure is a start: MSD notes that every calorie calculation is only a starting point, to be adjusted by how the animal responds [2]; body condition over the following weeks is the check.

Who this applies to

Healthy adult dogs, using the MSD Veterinary Manual's starting formulas. Not puppies beyond the multipliers given, not pregnant or nursing dogs, not dogs being treated for weight or illness.

Worked example the reader can copy

A 20 kg neutered dog: 20^0.75 = 9.46; x 70 = 662.0 kcal (RER); x 1.6 = 1059 kcal a day. Divide by the kcal per cup on the food bag to get cups.

What it cannot tell you

Your dog's activity, breed or body condition, or your food's calories. Every figure is a starting point.

Shape I would write

The answer: the formula and three weights.
The shortcut and where it drifts (the table), with the reason in one sentence.
From resting to daily: the multipliers.
Turning kcal into cups from the bag.
Check by body condition, not by the number.