Area, Perimeter & Radius of Circles & Polygons

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SUMMARY

The discussion centers on the mathematical relationships between the radius, area, and perimeter of regular polygons. It establishes that for regular polygons, the area can be calculated using the formula Area = (1/2) * number of sides * radius squared * sin(2π/n), where 'n' represents the number of sides. As the number of sides increases, the constants relating the radius to the area and perimeter approach the value of π (approximately 3.14). The conversation also explores the possibility of deriving these constants through equations involving the radius and apothem.

PREREQUISITES
  • Understanding of basic geometry concepts, including polygons and circles.
  • Familiarity with trigonometric functions, specifically sine.
  • Knowledge of mathematical constants, particularly π (pi).
  • Ability to manipulate algebraic equations involving variables.
NEXT STEPS
  • Research the derivation of the area formula for regular polygons using trigonometric identities.
  • Explore the relationship between the radius and apothem in regular polygons.
  • Study the concept of limits and how they apply to polygons with an increasing number of sides.
  • Investigate the mathematical properties of π and its relevance in geometry.
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Mathematicians, geometry enthusiasts, educators, and students seeking to deepen their understanding of the relationships between radius, area, and perimeter in polygons and circles.

scott_alexsk
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This has been bugging me for a while and I thought that you guys might know an answer. Awhile ago I realized that there is a direct relationship between the radius (as in the distance between a corner and the center) squared and the area of any regular polygon with the same number of sides. For example the radius of any square, squared and multiplied by 2 equals the area. But also, for any triangle the radius squared times approx. 1.3 equals the area.

By finding this for several polygons, I found what I expected, the constant for each polygon as the sides increase approaches 3.14. I also found that there is a constant between the diameter of a polygon and the perimeter which also approaches 3.14 as the number of sides increases on the regular polygon.

My question is, according to these relationships is there a way to determine the value of any of these constants for polygons including a circle (3.14) (assuming that it is a polygon with a infinite number of sides) using an equation? Is there any equation you can think of that shows this? Perhaps there is an equation that determines this from the relation of the radius and the apothum (sp)?

Thanks
-scott
 
Last edited:
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scott_alexsk said:
My question is, according to these relationships is there a way to determine the value of any of these constants for polygons including a circle (3.14) (assuming that it is a polygon with a infinite number of sides) using an equation? Is there any equation you can think of that shows this?
If the regular polygon has even no. of sides then
Area= (1/2)*no. of sides* side squared* sin(2pi/n)

Keep Smiling
Malay
 
That is also true when the polygon has an odd number of sides.

("side squared" means Scott's "radius" squared)
 
Area= (1/2)*no. of sides* radius squared* sin(2pi/n)

What's n stand for?

Thanks for all of the posts
-scott
 
scott_alexsk said:
Area= (1/2)*no. of sides* radius squared* sin(2pi/n)

What's n stand for?

Thanks for all of the posts
-scott
n stands for the number of sides.

Keep Smiling
Malay
 

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