Circumscribed and inscribed circles of a regular hexagon?

In summary, the ratio of the areas of the circumscribed and inscribed circles of a regular hexagon can be calculated using the formulas for the apotema and area of a regular polygon. The ratio is equal to the tangent of 30, or can be found using the Pythagorean theorem.
  • #1
josephcollins
59
0
Hey ppl,
Could anyone help me with this: what is the ratio of the areas of the circumscribed and inscribed circles of a regular hexagon? how do I go about working it out from first principles?
Cheers, joe
 
Mathematics news on Phys.org
  • #2
Well for any regular polygon its "apotema" (apothem perhaps?) is

[tex] a = \frac{L}{2}tan \frac{\alpha}{2} [/tex]

where [tex] L [/tex] is the length of a side and [tex] \alpha [/tex] is the inner angle of the polygon.

To calculate this inner angle just use the formula

[tex] \alpha = \frac{(n-2)180}{n} [/tex]

where n is the number of sides

and its Area is

[tex] A = \frac{1}{2}pa [/tex]

where p is the perimeter or nL

Now if i remember circumscribed correctly means a circle inside the hexagone and inscribed means a hexagone inside the circle, right?

maybe this could be calculated with right triangles...anyhow the formulas above could help you for a regular hexagone

Edit: sorry for so many edits, seems i need a break.
 
Last edited:
  • #3
The regular hexagon consists of six equilateral triangles, the radius of the inscribed circle is equal to the height (i.e. distance from top to middle of base) of one such triangle and the radius of the circumscribed circle is equal to the length of a side of one such triangle.
 
  • #4
In other words, the ratio of radii is clearly equal to the tangent of 30.
 
  • #5
Gokul43201 said:
In other words, the ratio of radii is clearly equal to the tangent of 30.

Or you just use phytagoras to see that:

heigth^2 + (1/2 * side)^2 = side^2
heigth^2 = side^2 - 1/4 * side^2
height^2 = 3/4 * side^2

and then:

area of circumscibed circle:
2Pi * side^2
area of inscribed circle:
2Pi * 3/4 * side^2

.
 

1. What is a circumscribed circle of a regular hexagon?

A circumscribed circle of a regular hexagon is a circle that passes through all six vertices of the hexagon, with the center of the circle located at the center of the hexagon.

2. How is the radius of a circumscribed circle of a regular hexagon calculated?

The radius of a circumscribed circle of a regular hexagon is equal to the length of any side of the hexagon. This can be calculated using the formula r = s/2, where r is the radius and s is the length of the side.

3. What is an inscribed circle of a regular hexagon?

An inscribed circle of a regular hexagon is a circle that is tangent to all six sides of the hexagon, with the center of the circle located at the center of the hexagon.

4. How is the radius of an inscribed circle of a regular hexagon calculated?

The radius of an inscribed circle of a regular hexagon is equal to half of the apothem, which is the distance from the center of the hexagon to any side. This can be calculated using the formula r = √3/2 * s, where r is the radius and s is the length of any side of the hexagon.

5. What is the relationship between the radii of the circumscribed and inscribed circles of a regular hexagon?

The radius of the circumscribed circle is always equal to twice the radius of the inscribed circle. In other words, the diameter of the circumscribed circle is equal to the apothem of the inscribed circle.

Similar threads

Replies
7
Views
5K
  • General Math
Replies
4
Views
4K
Replies
1
Views
7K
Replies
4
Views
3K
Replies
2
Views
2K
Replies
8
Views
2K
  • General Math
Replies
33
Views
2K
Replies
2
Views
997
  • Introductory Physics Homework Help
Replies
17
Views
1K
Back
Top