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.

Yes

Idk why you thought this was some kind of gotcha, it's just correct. It doesn't matter which one you use.

somehow 68°F being pants weather while 72°F is shorts weather makes more sense than interpreting 20°C vs. 22.2°C for the same decision.

why are you picking 20 vs 22.2 and not 21 vs 23 or 19 vs 22? still completely arbitrary. and shorts weather still depends on sunshine and humidity as much as pure temperature, a cloudy, wet 23 degrees is still cold, but a nice sunny 19 can be fine. i don't think anyone can actually feel the difference between 1°C, much less 1°F