Geometry

[Gopi]

Guys,

I have a query on the following problem, request you to look into this and let me know, whats the best way to tackle these kind of problems?


If a circle, regular hexagon and a regular octagon have the same area and if the perimeter of the circle is represented by "c", that of the hexagon by "h" and that of the octagon by "o", then which of the following is true?

A) c>o>h
B) c>h>o
C) h>c>o
D) o>h>c
E) h>o>c


Thanks
Gopi


My approach is bit time consuming and also is not yielding exact answer, forcing me to guess in the end.


Since Circle, Regular Hexagon and Regular Octagon are having equal areas.

¶R^2^2=6 √¾ H^2=8 √¾ O^2
(where, R-radius of the circle, H-is the side of the hexagon and O-is the side of the octagon)

3.14R^2=2.55H^2=3.4O^2

6h, 2¶R and 8o
(regular hexagon, a circle and a regular octagon respectively)

from here on, the answer is not getting any clearer so i had to guess and move on.

So guys, look into this and provide any better and quicker way to deal this problem.

[gauravgundal]

IMO A



Area of circle Area of Octagon Area of Hexagon
A = Pie (radius)^2 = (2+2sqrt2)side^2 = ((3sqrt3)/2)side^2

get the value of radius in terms of A same for OCtagon and hexagon

substitute those value in circumference of their respective.

[ket]

IMO A



Area of circle Area of Octagon Area of Hexagon
A = Pie (radius)^2 = (2+2sqrt2)side^2 = ((3sqrt3)/2)side^2

get the value of radius in terms of A same for OCtagon and hexagon

substitute those value in circumference of their respective.

gauravgundal could you please explain in more detail? I am not getting your explanation.

a. why is area of octagon (2+2sqrt2)side^2 - I tried it this way regular octagon is 8 times the 8 regular triangles that could be drawn in it. let the side of octagon be B than the area of octagon is: 8 X ( B^2 X (3)^1/2 X 1/4 )

b. I also dont understand why u used 'side' for both octagon and hexagon, we can not imply that sides are equal right?

Thanks

[tohellandback]

if perimeter is same
area of circle>area of octagon>area of hexagon

so what I figure is
if area is same

perimeter of hexagon>perimeter of octagon>perimeter of circle.

so E
but I am not sure.
what's the OA?

[Gopi]

Guys,

Looking for a detailed explanation and a possible shortcut.


The answer is D.

Thanks
Gopi

[ket]

On second thought I think it's D.

See attached file for solution.

[maihuna]

IMO A



a. why is area of octagon (2+2sqrt2)side^2 - I tried it this way regular octagon is 8 times the 8 regular triangles that could be drawn in it. let the side of octagon be B than the area of octagon is: 8 X ( B^2 X (3)^1/2 X 1/4 )


Thanks

Gopya, be careful a octagon is not a hexagon, you have 8 triangles while the total angle at centre is 360 so each triangle will have a vertex angle of 360/8 = 45

Since the two sides joining the centre with the octagon vertex will be equal it will be an isosceles triangle with remaining 135 divided twice, so you are dealing with an isosceles triangle.

[maihuna]

On second thought I think it's D.

See attached file for solution.

On third thought I think you did miserably wrong, how can the area of octagon be 8\/3b^2/4

Do you think there will be eight equilateral triangles wow....how man...8*60 = 480 you have only 360 in a euclid geom...

[ket]

thanks a lot maihuna.. I made a mistake about Octagon

I didn't even know the area formula for Octagon... thats too much for GMAT don't you think? :) I think this kind of problems don't show up in real tests...

[maihuna]

Guys,

Gopya a very interesting feature of regular shapes is in test here, for a given perimeter circle is the one that has maximum area, and what is a circle, a polygon of maximum side what does it mean, given a perimeter triangle will have lesser area than square, square is less than pentagon, pentagon less than hexagon, bla bla, finally octagon greater than hexagon..

see it this way: for a perimeter 36, side of triangle is 12, square is 9
area of triangle: \/3/4*12^2 = (144/4)*1.73 = 38*1.73 = 65.74
area of square: 9^2 = 81

similarly you may check for other things... since area is equal the perimeters will follow the suit, i.e. circle will have lowest perimeter followed by octagon followed by hexagon...

go on choose option E w/o fear. you will get it right...dont do any cal that is not tested here....