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{{Short description|Minimum distance from the center to the side of a regular polygon}}
== Armstrong Limit ==


The '''apothem''' of a regular polygon is the distance from the center to the midpoint of one of its sides. It is a key concept in geometry, particularly in the study of regular polygons, and is used in various calculations, including the area of the polygon.
The '''Armstrong limit''', also known as '''Armstrong's line''', is the altitude above which atmospheric pressure is sufficiently low that water boils at the normal temperature of the human body, approximately 37 °C (98.6 °F). This limit is named after [[Harry George Armstrong]], who was a pioneering figure in aviation medicine.


==Definition==
== Atmospheric Pressure and Boiling Point ==
In a regular polygon, all sides and angles are equal. The apothem is the perpendicular distance from the center of the polygon to one of its sides. It can also be considered as the radius of the inscribed circle (incircle) of the polygon.


==Properties==
[[File:Comparison_International_Standard_Atmosphere_space_diving.svg|Comparison of International Standard Atmosphere for space diving|thumb|right]]
The apothem is an important element in determining the area of a regular polygon. The formula for the area \(A\) of a regular polygon with \(n\) sides, each of length \(s\), and apothem \(a\) is given by:


\[
At sea level, the atmospheric pressure is about 101.3 kPa (14.7 psi), which is sufficient to keep water in a liquid state at body temperature. However, as altitude increases, atmospheric pressure decreases. At the Armstrong limit, which is approximately 18,900 meters (62,000 feet) above sea level, the pressure drops to about 6.3 kPa (0.91 psi). At this pressure, the boiling point of water matches the normal body temperature.
A = \frac{1}{2} \times n \times s \times a
\]


This formula can also be expressed in terms of the perimeter \(P\) of the polygon:
== Implications for Human Physiology ==


\[
Above the Armstrong limit, humans cannot survive without a pressurized environment. If exposed to such low pressures, the moisture in the lungs and other body tissues would begin to boil, leading to a condition known as ebullism. This is a serious medical emergency that can result in rapid loss of consciousness and death if not addressed immediately.
A = \frac{1}{2} \times P \times a
\]


==Calculation==
[[File:F-16_pilot,_closeup,_canopy_blemishes_cleaned.jpg|F-16 pilot closeup|thumb|left]]
The apothem can be calculated if the side length \(s\) and the number of sides \(n\) of the regular polygon are known. The formula for the apothem \(a\) is:


\[
To prevent ebullism and other altitude-related health issues, pilots and astronauts use pressurized suits or cabins. These environments maintain a pressure similar to that at lower altitudes, allowing the body to function normally.
a = \frac{s}{2 \tan(\pi/n)}
\]


This formula arises from the fact that the apothem is the adjacent side of a right triangle formed by the radius of the circumscribed circle, the apothem itself, and half of a side of the polygon.
== Historical Context and Research ==


==Applications==
The concept of the Armstrong limit was first identified in the 1940s by Harry George Armstrong, who was studying the effects of high-altitude flight on human physiology. His research laid the groundwork for modern aviation and space medicine.
The apothem is used in various applications, including:


* Calculating the area of regular polygons.
[[File:Caproni_Ca.161_pilot.jpg|Caproni Ca.161 pilot|thumb|right]]
* Determining the radius of the inscribed circle.
* Architectural design and construction, where regular polygons are used in tiling and other patterns.


==Examples==
== Applications in Aviation and Space Exploration ==
For a regular hexagon with side length \(s\), the apothem can be calculated using the formula:


\[
The Armstrong limit is a critical consideration in the design of high-altitude aircraft and spacecraft. For example, pilots of high-altitude reconnaissance aircraft, such as the [[Lockheed U-2]], must wear pressure suits to protect against the low-pressure environment.
a = \frac{s}{2 \tan(\pi/6)} = \frac{s}{\sqrt{3}}
\]


This is because a regular hexagon can be divided into six equilateral triangles, and the apothem is the height of one of these triangles.
[[File:High-altitude_balloon_-_ATLAS0_mission.jpg|High-altitude balloon during ATLAS0 mission|thumb|left]]


==Related pages==
In space exploration, the Armstrong limit is surpassed by the vacuum of space, where pressure is effectively zero. Spacecraft and space suits are designed to provide a pressurized environment to ensure the safety and survival of astronauts.
* [[Regular polygon]]
* [[Circumscribed circle]]
* [[Inscribed circle]]
* [[Perimeter]]


==Gallery==
== Related Pages ==
<gallery>
File:Apothem_of_hexagon.svg|Diagram showing the apothem of a hexagon
</gallery>


[[Category:Geometry]]
* [[Aviation medicine]]
* [[Space medicine]]
* [[Ebullism]]
* [[Pressure suit]]
* [[High-altitude flight]]
 
[[Category:Aviation medicine]]
[[Category:Atmospheric science]]
[[Category:Human physiology]]

Latest revision as of 11:19, 23 March 2025

Armstrong Limit[edit]

The Armstrong limit, also known as Armstrong's line, is the altitude above which atmospheric pressure is sufficiently low that water boils at the normal temperature of the human body, approximately 37 °C (98.6 °F). This limit is named after Harry George Armstrong, who was a pioneering figure in aviation medicine.

Atmospheric Pressure and Boiling Point[edit]

Comparison of International Standard Atmosphere for space diving

At sea level, the atmospheric pressure is about 101.3 kPa (14.7 psi), which is sufficient to keep water in a liquid state at body temperature. However, as altitude increases, atmospheric pressure decreases. At the Armstrong limit, which is approximately 18,900 meters (62,000 feet) above sea level, the pressure drops to about 6.3 kPa (0.91 psi). At this pressure, the boiling point of water matches the normal body temperature.

Implications for Human Physiology[edit]

Above the Armstrong limit, humans cannot survive without a pressurized environment. If exposed to such low pressures, the moisture in the lungs and other body tissues would begin to boil, leading to a condition known as ebullism. This is a serious medical emergency that can result in rapid loss of consciousness and death if not addressed immediately.

F-16 pilot closeup

To prevent ebullism and other altitude-related health issues, pilots and astronauts use pressurized suits or cabins. These environments maintain a pressure similar to that at lower altitudes, allowing the body to function normally.

Historical Context and Research[edit]

The concept of the Armstrong limit was first identified in the 1940s by Harry George Armstrong, who was studying the effects of high-altitude flight on human physiology. His research laid the groundwork for modern aviation and space medicine.

Caproni Ca.161 pilot

Applications in Aviation and Space Exploration[edit]

The Armstrong limit is a critical consideration in the design of high-altitude aircraft and spacecraft. For example, pilots of high-altitude reconnaissance aircraft, such as the Lockheed U-2, must wear pressure suits to protect against the low-pressure environment.

High-altitude balloon during ATLAS0 mission

In space exploration, the Armstrong limit is surpassed by the vacuum of space, where pressure is effectively zero. Spacecraft and space suits are designed to provide a pressurized environment to ensure the safety and survival of astronauts.

Related Pages[edit]