Editing 2748: Radians Are Cursed

{{comic
| number    = 2748
| date      = March 10, 2023
| title     = Radians Are Cursed
| image     = radians_are_cursed_2x.png
| imagesize = 394x437px
| noexpand  = true
| titletext = Phil Plait once pointed out that you can calculate the total angular area of the sky this way. If the sky is a sphere with radius 57.3 degrees, then its area is 4*pi*r^2=41,253 square degrees. This makes dimensional analysts SO mad, but you can't argue with results.
}}

==Explanation==
This comic presents a series of Math Facts, appearing to be in a sequential order.

The first fact states that the {{w|unit circle}} has a radius of one, which is precisely its definition. Randall labels this fact as being "normal," complete with a large green checkmark to verify this. The unit circle is typically used in abstract contexts rather than applications with a specific length unit (such as meters), leading to the "unit" being treated as dimensionless. For example, the dimensionless trigonometric functions cosine and sine can be defined as the x and y coordinates of a point on the unit circle.

The second fact states that one {{w|radian}} is equal to the length of a circle's radius. This isn't actually the way that the unit is defined. Instead, radians are usually defined as the angle encompassing an arc of a circle equal in length to its radius. This comic's logic is thus somewhat erroneous. However, this fact is still labeled as also being "normal." Also, while Randall again uses the unit circle in the fact's associated diagram, any circle could theoretically be used to show the conventional definition. Under the standard definition of an {{w|angle}} as the ratio of the length of a circular arc to its radius, the radian is a dimensionless unit equal to 1. 

A correct version of the second fact would be that a radian has the same value (1) as the radius of the unit circle.

The third fact states that one radian is equal to 57.3°. This is indeed true (albeit rounded). The circumference of a circle is 2π radius-lengths, so the angle of a complete circle is both 2π rad and 360°. Thus 1 rad = 180/π°. This fact is again labeled "normal."

The fourth and final fact states that because it was determined in earlier facts that a radian is equal to the radius of the unit circle as well as 57.3°, then the radius of the unit circle must be equal to 57.3°. This is usually not how degrees are supposed to work, because they are a measure of angle, not length. Hence, this fact is labeled "cursed" by Randall, leading to the comic's title. (However, since the radian is also an angular measure, the second fact could be viewed as equally cursed.)

The title text is referring to {{w|Phil Plait}}'s claim about the size of the sky, which was published on his blog:  http://www.badastronomy.com/bitesize/bigsky.html. Dimensional analysis utilizes the rationale that both sides of an equation need to have the same unit. Radius typically refers to a length, which has SI units of meters. The surface area has SI units of square meters. The units of Phil Plait's "angular area" is as the title text mentions, {{w|square degrees}}. Thus the comic's {{w|dimensional analysis|dimensional analysts}} (not a profession, but instead the adherents of the mathematical technique) are said to be angered by this argument.

Randall has alluded to Plait's angular area tip previously [https://what-if.xkcd.com/109/ in his own blog What If?], in a post that examined the chances of hitting various celestial objects with a laser blast aimed at random from Earth's surface.

==Transcript==
:[Title:]
:<u>Math facts</u>

:[A diagram of a circle is shown with radius labeled as “1”]
:The unit circle has a radius of one [In green] <span style="color:#008000;">✓ Normal</span>
:[The diagram now has another (unlabeled and lighter) radius at a 57.3 degree angle. The arc between the points where the radii intersect the circle is labeled as “1”]
:One radian equals the length of a circle’s radius [In green] <span style="color:#008000;">✓ Normal</span>
:[The diagram now is completely unlabeled except for the arc, which is labeled as “57.3°”]
:One radian is 57.3 degrees [In green] <span style="color:#008000;">✓ Normal</span>
:[The first diagram is shown; however, the radius is labeled as “57.3°”]
:The unit circle has a radius of 57.3 degrees [In red] <span style="color:#800000;">✗ Cursed</span>

{{comic discussion}}

[[Category:Math]]
[[Category:Geometry]]
[[Category:Facts]]
[[Category:Comics with color]]
[[Category:Comics featuring real people]] <!-- Phil Plait -->
[[Category:Comics with cursed items]]