Question

a regular polygon A has the midpoints of its edges joined to form a smaller hexagon B to get a third hexagon C. What is the ratio of the area of hexagon C to the area of hexagon A?

Answer 1

equation for area of a hexagom =...?

Answer 2

Area of Polygon = ¼ × n × (Side length)^2 / tan(π/n)
n = number of sides = 6 for a hexagon

so:
\[Area.of.Hexagon = \frac 14 × 6 × (side)^2 / \tan(π/6)\]

Answer 3

\[Area.of.Hexagon = 1.5 × (side)^2 / \tan(π/6)\]
now tan (pi/6) = 1/sqrt 3
so
\[Area.of.Hexagon = 1.5 × \sqrt 3 × (side)^2 \]

Answer 4

ratio of side length = 2:1
new sidelength is 1/2 original side length
so new area is (1/2)^2=1/4
so do that again
new side length = 1/4 of original
so new area is (1/4)^2=1/16 th area of original area

Answer 5

Similar problem:

http://openstudy.com/study#/updates/507d2c0ae4b040c161a2bc42

Answer 6

Suppose The Length Of One Side Of The Hexagon A Is 1cm
(Interior Angle Of A Regular Hexagon Is 120 Degrees).
Midpoint Would Be 0.5 cm Away From The Edge.
Then for Hexagon B, The Length Of One Side
Using The Cosine Rule.
=>Cos120 = (0.5^2 + 0.5^2 -c^2)/2*0.5*0.5
=>c = 0.866cm
The Midpoint For ""B"" Would Be 0.866/2 cm
From The Edge Of ""B"" which is 0.433cm
Same For Hexagon C
=>Cos120 = (0.433^2 + 0.433^2 -x^2)/2*0.433*0.433
=>x = 0.75 cm
The Hexagons (A, B & C) Are Geometrically Similar
we have
=>(length1/length2)^2 = Area1/Area2
hence
=>(0.75/1)^2 = Area C/Area A
we have 1cm was the length of one side of A and 0.75 was the length of one side of C
=>Area C/ Area A = 0.5625 = 9/16
ratio of the area of hexagon C to the area of hexagon A =9/16

Know more about Perimeter of a Polygon http://goo.gl/HN2M6u