It is πr^2^ or, equivalently, 1/2 τr^2^
But wait since π is defined in terms of the diameter it should be 1/4 πD^2^ .
In fact look at all these quadratic forms
- distance fallen in a gravitational field: 1/2 gt^2^
- energy of motion: 1/2 mv^2^
- speed 1/2 at^2^
- area of a circle 1/2 τr^2^
Because when you step up from the linear, one dimensional circumference, and integrate, then the antiderivative should have a 1/2 factor to account for the square when differentiating back. The fact that π cancels this factor, and hides it is to its detriment!
Read https://www.tauday.com/tau-manifesto again, specifically section 3. τ is how the circumference of a unit circle should be defined since we use the radius and never the diameter. Anything else is revisionism comrade, do better.