area of shaded region
This topic has expert replies
-
- Legendary Member
- Posts: 829
- Joined: Mon Jul 07, 2008 10:09 pm
- Location: INDIA
- Thanked: 84 times
- Followed by:3 members
Another question that stumpped me !!!
- Attachments
-
- hexagon.doc
- (92.5 KiB) Downloaded 126 times
-
- Master | Next Rank: 500 Posts
- Posts: 222
- Joined: Fri Jan 04, 2008 8:10 pm
- Thanked: 15 times
Find the area of hexagon
it has 6 equilateral triangles
area of equilateral triangle is =sqrt(3)/4*side^2
=sqrt(3)/4*6^2
=sqrt(3)/4*36
=sqrt(3)*9
for six equilateral triangles =6*sqrt(3)*9
=54sqrt(3)----(1)
area of circle is =pi *r^2
=pi*3^2
=9pi-----(2)
area of portion of the circle is 120/720* 9pi =3/*pi
then area of all portions =9pi----(3)
(1)-(2)-(3)
gives 54sqrt(3)-9pi-9pi
C
it has 6 equilateral triangles
area of equilateral triangle is =sqrt(3)/4*side^2
=sqrt(3)/4*6^2
=sqrt(3)/4*36
=sqrt(3)*9
for six equilateral triangles =6*sqrt(3)*9
=54sqrt(3)----(1)
area of circle is =pi *r^2
=pi*3^2
=9pi-----(2)
area of portion of the circle is 120/720* 9pi =3/*pi
then area of all portions =9pi----(3)
(1)-(2)-(3)
gives 54sqrt(3)-9pi-9pi
C
Last edited by sibbineni on Sun Aug 10, 2008 11:42 am, edited 1 time in total.
-
- Legendary Member
- Posts: 1153
- Joined: Wed Jun 20, 2007 6:21 am
- Thanked: 146 times
- Followed by:2 members
I think the answer C, not too sure coz I make lot of silly mistakes.
The sum of all the angles = (n-2)*180 = 4*180 = 720
perimeter = 36, therefore the length of AB, BC, CD,DE, EF, FA=6
draw a line from each of the points ABCDEF to the center, you will have 6 equilateral triangles.
area of first triangle = a^2sqrt3/4= 36sqrt3/4= 9sqrt3
area of the hexagon = 6*(9sqrt3) = 54sqrt3
since all sides of the hexagon are equal, all the angles will also be equal
Therefore 720/6 = 120 is each individual central angle.
area of portion of the circle = 120/720 * 9pi = 3/2 pi
area of all the portions of the circle = 3/2pi * 6 = 9 pi
area of the circle inside = 9pi
area of the shaded region = 54sqrt3 -9pi-9pi = 54sqrt3 -18pi.
Hence C is the answer.
Whats the OA?
The sum of all the angles = (n-2)*180 = 4*180 = 720
perimeter = 36, therefore the length of AB, BC, CD,DE, EF, FA=6
draw a line from each of the points ABCDEF to the center, you will have 6 equilateral triangles.
area of first triangle = a^2sqrt3/4= 36sqrt3/4= 9sqrt3
area of the hexagon = 6*(9sqrt3) = 54sqrt3
since all sides of the hexagon are equal, all the angles will also be equal
Therefore 720/6 = 120 is each individual central angle.
area of portion of the circle = 120/720 * 9pi = 3/2 pi
area of all the portions of the circle = 3/2pi * 6 = 9 pi
area of the circle inside = 9pi
area of the shaded region = 54sqrt3 -9pi-9pi = 54sqrt3 -18pi.
Hence C is the answer.
Whats the OA?
sibbineni wrote:Find the area of hexagon
it has 6 equilateral triangles
area of equilateral triangle is =sqrt(3)/4*side^2
=sqrt(3)/4*6^2
=sqrt(3)/4*36
=sqrt(3)*9
for six equilateral triangles =6*sqrt(3)*9
=54sqrt(3)----(1)
area of circle is =pi *r^2
=pi*3^2
=9pi-----(2)
area of portion of the circle is 120/720* 9pi =3/*pi
then area of all portions =9pi----(3)
(1)-(2)-(3)
gives 54sqrt(3)-9pi-9pi
C
I did the same exact up until the 3rd part. why do you divide 120/720? shouldn't it be 120/360 which is the portion of the circle and multiply by the area?
I have 120/360 x 9pi = 3pi x # of portions (6) = 18pi
Thus, 54sqr3-9pi-18pi = 54sqr3-27pi (E)
What do you think?
-
- Legendary Member
- Posts: 1153
- Joined: Wed Jun 20, 2007 6:21 am
- Thanked: 146 times
- Followed by:2 members
Good work Kshin78. Absolutely Perfect.
Cant argue with that. Answer has to be E.
Like I said in my post I make lot of silly mistakes.
Cant argue with that. Answer has to be E.
Like I said in my post I make lot of silly mistakes.
-
- Legendary Member
- Posts: 661
- Joined: Tue Jul 08, 2008 12:58 pm
- Location: France
- Thanked: 48 times
Let's find the areas of the 7 circles.
The radius of the circles is 3 given side of the hegaxon is 6 and we have to radius for each side.
We have one total circle so its area is 9pi
The area of the other 6 circles is 120/360*9pi*6=18pi (proportion of the circle (we know each angle of an hexagon is 120° and a circle is 360°) times the area of each circle times the number of circles)
Now the hegaxon:
The area of the hegaxon is 3sqrt(3)/2 * 6^2 = 54sqrt(3)
(General formula for regular hegaxon is 3sqrt(3)/2*a^2 with a one side, useful but hard to remember)
Now we can subtract all that and we have 54sqrt(3)-27pi
That is a nice question
The radius of the circles is 3 given side of the hegaxon is 6 and we have to radius for each side.
We have one total circle so its area is 9pi
The area of the other 6 circles is 120/360*9pi*6=18pi (proportion of the circle (we know each angle of an hexagon is 120° and a circle is 360°) times the area of each circle times the number of circles)
Now the hegaxon:
The area of the hegaxon is 3sqrt(3)/2 * 6^2 = 54sqrt(3)
(General formula for regular hegaxon is 3sqrt(3)/2*a^2 with a one side, useful but hard to remember)
Now we can subtract all that and we have 54sqrt(3)-27pi
That is a nice question
-
- Legendary Member
- Posts: 829
- Joined: Mon Jul 07, 2008 10:09 pm
- Location: INDIA
- Thanked: 84 times
- Followed by:3 members