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Area and Perimeter of Rectangles

Area is base times height, while perimeter is the sum of the sides.

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The Infinite Coastline Paradox

Credit: Barry Lewis
Source: http://www.flickr.com/photos/16179216@N07/7159496260/
License: CC BY-NC 3.0

The length of the coastline of Great Britain is 17,820 kilometers… or is it? Some might argue that coastline lengths are actually infinite! How is that possible? It all depends on how you choose to measure.

Amazing But True

Coastlines have extremely jagged edges formed by the constant and powerful eroding forces of wind, water, ice, and gravity. This makes measuring the perimeter of a real landmass much tougher than measuring the perimeter of a straight-edged polygon. Imagine measuring the length of a coastline. You start with a kilometer-long instrument. Even if you adjust the instrument at each kilometer to follow the direction of the coast, you will still overlook many smaller jagged outcrops. You decide you want more detail, so you begin measuring with a meter stick, again adjusting every meter to account for as many details in the coastline as possible. You would still miss some jagged details between meters, and this second coastline measurement would be longer than your first! You decide to get even more precise and measure with a centimeter stick, then a millimeter stick, and so on. Each coastline measurement will get longer as you make your instrument smaller.

Credit: Laura Guerin
Source: CK-12 Foundation
License: CC BY-NC 3.0

Some would call this the "infinite coastline paradox." The length of a coastline appears to be much smaller when measured by kilometers than it does when measured by millimeters. Actually, measuring with an infinitely small instrument would result in an infinitely long coastline. When we talk about infinite geometric patterns (like the edges of a landmass) that get smaller and more detailed the further you “zoom in,” we’re talking about fractal geometry.

See for yourself in this video about the Australian coastline and fractals:

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Read the following article to find out more about the history of the coastline paradox, and learn more about fractals and self-similarity in nature with the CK-12 Concept linked below.



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Image Attributions

  1. [1]^ Credit: Barry Lewis; Source: http://www.flickr.com/photos/16179216@N07/7159496260/; License: CC BY-NC 3.0
  2. [2]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0

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