explain xkcd - User contributions [en]

2748: Radians Are Cursed

2023-03-11T09:52:59Z

Aseyhe: /* Explanation */

{{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==
{{incomplete|Created by a DIAMETER OF 114.6 DEGREES - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

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 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 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 degrees. This is indeed true (albeit rounded). The circumference of a circle is 2pi radius-lengths, so the angle of a complete circle is both 2pi radians and 360 degrees. Thus 1 radian equals 180/pi degrees. 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 degrees, then the radius of the unit circle must be equal to 57.3 degrees. 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==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Title:]
:Math facts

:[A diagram of a circle is shown with radius labeled as “1”]
:The unit circle has a radius of one  ✓ Normal
:[The diagram now has another (unlabeled and lighter) radius at a 57.3 degree angle. The arc between the points where the radii touch the circle is labeled as “1”]
:One radian equals the length of a circle’s radius  ✓ Normal
:[The diagram now is completely unlabeled except for the arc, which is labeled as “57.3°”]
:One radian is 57.3 degrees  ✓ Normal
:[The first diagram is shown; however, the radius is labeled as “57.3°”]
:The unit circle has a radius of 57.3 degrees  X Cursed

{{comic discussion}}

[[Category:Math]]
[[Category:Geometry]]
[[Category:Facts]]
[[Category:Comics featuring real people]]

2748: Radians Are Cursed

2023-03-10T21:35:45Z

Aseyhe: /* Explanation */

{{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==
{{incomplete|Created by a CURSED RADIAN - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

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

A correct version of the second fact would be that, 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. Thus, by the first fact, a radian is equal to the radius of the unit circle.

The third fact states that one radian is equal to 57.3 degrees. The more usual conversion factor between the two units is that pi radians is equal to 180 degrees, though the conversion given by Randall is still indeed true (albeit rounded). This 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 degrees, then the radius of the unit circle must be equal to 57.3 degrees. 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 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==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Title:]
:Math facts

:[A diagram of a circle is shown with radius labeled as “1”]
:The unit circle has a radius of one  ✓ Normal
:[The diagram now has another (unlabeled and lighter) radius at a 57.3 degree angle. The arc between the points where the radii touch the circle is labeled as “1”]
:One radian equals the length of a circle’s radius  ✓ Normal
:[The diagram now is completely unlabeled except for the arc, which is labeled as “57.3°”]
:One radian is 57.3 degrees  ✓ Normal
:[The first diagram is shown; however, the radius is labeled as “57.3°”]
:The unit circle has a radius of 57.3 degrees  X Cursed

{{comic discussion}}

[[Category:Math]]
[[Category:Geometry]]
[[Category:Facts]]
[[Category:Comics featuring real people]]

2748: Radians Are Cursed

2023-03-10T21:18:57Z

Aseyhe: /* Explanation */

{{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==
{{incomplete|Created by a CURSED RADIAN - Please change this comment when editing this page. Do NOT delete this tag too soon.}}

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

A correct version of the second fact would be that, 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. Thus, by the first fact, a radian is equal to the radius of the unit circle.

The third fact states that one radian is equal to 57.3 degrees. The more usual conversion factor between the two units is that pi radians is equal to 180 degrees, though the conversion given by Randall is still indeed true (albeit rounded). This is again labeled "normal."

The fourth and final fact, however, is a questionable application of the {{w|transitive property}}, stating that because it was determined in earlier facts that a radian is equal to the radius of the unit circle (still not entirely correct) as well as 57.3 degrees, then the radius of the unit circle must be equal to 57.3 degrees. This is usually not how degrees are supposed to work in mathematics, and is thus the only fact labeled "cursed" by Randall, hence the title of the comic.

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, square degrees, a unit not used in any scientific unit system. Thus the comic's {{w|dimensional analysis|dimensional analysts}} (not a profession, but instead the adherents of the mathematical technique) are made quite angry by the application of Randall's strange and unconventional logic.

Randall has alluded to Plait's angular area tip previously 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==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[Title:]
:Math facts

:[A diagram of a circle is shown with radius labeled as “1”]
:The unit circle has a radius of one  ✓ Normal
:[The diagram now has another (unlabeled and lighter) radius at a 57.3 degree angle. The arc between the points where the radii touch the circle is labeled as “1”]
:One radian equals the length of a circle’s radius  ✓ Normal
:[The diagram now is completely unlabeled except for the arc, which is labeled as “57.3°”]
:One radian is 57.3 degrees  ✓ Normal
:[The first diagram is shown; however, the radius is labeled as “57.3°”]
:The unit circle has a radius of 57.3 degrees  X Cursed

{{comic discussion}}

[[Category:Math]]
[[Category:Geometry]]
[[Category:Facts]]
[[Category:Comics featuring real people]]

Talk:2748: Radians Are Cursed

2023-03-10T20:57:24Z

Aseyhe:

how do transcript [[Special:Contributions/172.70.127.37|172.70.127.37]] 19:23, 10 March 2023 (UTC)

https://en.wikipedia.org/wiki/Square_degree may be of some help with this one. [[Special:Contributions/162.158.166.124|162.158.166.124]] 19:44, 10 March 2023 (UTC)

The comic isn't actually correct. A radian is not equal to the length of a circle's radius; it is equal to the length of the radius, multiplied by 2π, divided by the perimeter, which is why it has no units, while the length does. In other words, radian/2pi=length of radius/length of perimeter. [[Special:Contributions/172.70.46.84|172.70.46.84]] 19:51, 10 March 2023 (UTC)

As suggested by the above Wikipedia link, square degrees are in fact often used in astronomical contexts. Also, it's quite standard to say that radian=1; see for example [https://en.wikipedia.org/wiki/SI_derived_unit SI derived unit]. An angle is the ratio between the arc length and the radius, and we just optionally append "radian" for clarity. So 1 = 57.3 degrees is correct; Randall simply used the wrong argument to obtain it. [[User:Aseyhe|Aseyhe]] ([[User talk:Aseyhe|talk]]) 20:57, 10 March 2023 (UTC)