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Director  Joined: 29 Nov 2012
Posts: 695
If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 74% (01:29) correct 26% (01:55) wrong based on 242 sessions

HideShow timer Statistics If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. $$12 \sqrt{3}$$

B. $$8 \pi$$

C. $$18 \sqrt{2}$$

D. $$24 \sqrt{3}$$

E. 48

Attachment: screen_shot_2010_12_23_at_3.41.13_pm.png [ 23.63 KiB | Viewed 177758 times ]

Originally posted by fozzzy on 02 Jun 2013, 01:38.
Last edited by Bunuel on 20 Feb 2019, 00:47, edited 3 times in total.
Renamed the topic and edited the question.
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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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In an regular hexagon inscribed in a circle, its side is equal the radius.
We can divide the hexagon in 6 triangles each with the base of 4. The heigth will equal $$\sqrt{4^2-2^2}=\sqrt{12}=2\sqrt{3}$$. To obtain this just use Pythagoras, the hypotenuse of each triangle it's the radius, and the bases it's $$\frac{4}{2}=2$$.

Now you have the height of each triangle, so $$A_t=(4*2\sqrt{3})/2=4\sqrt{3}$$.

$$A_h=6*A_t=6*4\sqrt{3}=24\sqrt{3}$$
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Posts: 113
Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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fozzzy wrote:
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

a) 12 root 3
b) 8 pi
c) 18 root 2
d) 24 root 3
e) 48

Image
https://s3.amazonaws.com/production.gro ... .13_pm.png

Regular hexagon will have six traingle regions. Central angle will be 60. (360/6)
Equilateral triangle will be formed as we have three 60 angles. Length of each side is equal to radius 4.
Area of hexagon = area of 6 eq. triangles = 6 * root 3/4 * 4^2 = 24 root3
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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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fozzzy wrote:
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

a) 12 root 3
b) 8 pi
c) 18 root 2
d) 24 root 3
e) 48

Image
https://s3.amazonaws.com/production.gro ... .13_pm.png

A regular hexagon is essentially composed of 6 equilateral trianlges...and the line joining the opposite vertices is the diameter of the circle in which the hexagon is inscribed...So the radius of the circle forms the side of the equilateral triangle...
Area is 6* (3^1/2)/4(a^2) where a = radius of the circle.
6*sq. rt3/4 * 4^2=24 root 3
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Joined: 02 Sep 2009
Posts: 58445
Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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fozzzy wrote:
Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. 12 root 3
B. 8 pi
C. 18 root 2
D. 24 root 3
E. 48

a-regular-hexagon-has-a-perimeter-of-30-units-what-is-the-131591.html
regular-hexagon-abcdef-has-a-perimeter-of-36-o-is-the-cente-89544.html

Hope it helps.
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Posts: 16
Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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1
fozzzy wrote:
Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. 12 root 3
B. 8 pi
C. 18 root 2
D. 24 root 3
E. 48

The Hexagon can be divided into 6 equilateral triangles , each with side = radius of the circle. Since area of equilateral triangle is (root(3)*a^2)4, for 6 such triangles we will get 24root3
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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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3
1) Each of the hexagon's angles 120 degrees : formula if you don't know it is [(# of sides - 2) x 180] = (6 - 2) x 180 = 720 ÷ 6 = 120.
2) Next, split up the hexagon into 6 equilateral triangles with 4 for each of its sides.
3) Find the area of one of the triangles:
- Base = 4
- find the height by splitting the triangle in half so that it becomes a 30/60/90 triangle and find the height using the pythagorean theorem or knowing the 1,√3,2 triangle. Height = 2√3
- one triangle's area = (1/2)bh = (1/2)(4)(2√3) = 4√3
4) find the area of the hexagon by multiplying the one triangle's area by 6:
- 6 x 4√3 = 24√3
Director  Joined: 29 Nov 2012
Posts: 695
Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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lchen wrote:
1) Each of the hexagon's angles 120 degrees : formula if you don't know it is [(# of sides - 2) x 180] = (6 - 2) x 180 = 720 ÷ 6 = 120.
2) Next, split up the hexagon into 6 equilateral triangles with 4 for each of its sides.
3) Find the area of one of the triangles:
- Base = 4
- find the height by splitting the triangle in half so that it becomes a 30/60/90 triangle and find the height using the pythagorean theorem or knowing the 1,√3,2 triangle. Height = 2√3
- one triangle's area = (1/2)bh = (1/2)(4)(2√3) = 4√3
4) find the area of the hexagon by multiplying the one triangle's area by 6:
- 6 x 4√3 = 24√3

Great method we could use the formula for the area of equilateral triangle and save a few steps Root 3/4 a^2 where a = 4
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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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fozzzy wrote:
Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. 12 root 3
B. 8 pi
C. 18 root 2
D. 24 root 3
E. 48

the hexagon comprises 6 identical equilateral triangles
area of 1 equilateral triangle=√3/4*4^2=4√3
6*4√3=24√3
D
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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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fozzzy wrote:
Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. 12 root 3
B. 8 pi
C. 18 root 2
D. 24 root 3
E. 48

We may recall that a regular hexagon can be divided into 6 equilateral triangles, and thus the area of a regular hexagon is:

6[(s^2√3)/4], where s = side of the equilateral triangle and (s^2√3)/4 = the area of the equilateral triangle.
Since the radius of 4 also represents one side of the equilateral triangle, we can now determine the area of the hexagon.

Area = 6[(4^2√3)/4]

Area = 6[(16√3)/4]

Area = 6(4√3)

Area = 24√3

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Joined: 22 Apr 2019
Posts: 31
Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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Bunuel wrote:
fozzzy wrote:
Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. 12 root 3
B. 8 pi
C. 18 root 2
D. 24 root 3
E. 48

http://gmatclub.com/forum/a-regular-hex ... 31591.html
http://gmatclub.com/forum/regular-hexag ... 89544.html

Hope it helps.

Will the length of the side of a regular polygon (not just a hexagon) inscribed in a circle always be equal to the radius?
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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Tactic101 wrote:
Bunuel wrote:
fozzzy wrote:
Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png
If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. 12 root 3
B. 8 pi
C. 18 root 2
D. 24 root 3
E. 48

http://gmatclub.com/forum/a-regular-hex ... 31591.html
http://gmatclub.com/forum/regular-hexag ... 89544.html

Hope it helps.

Will the length of the side of a regular polygon (not just a hexagon) inscribed in a circle always be equal to the radius?

No. Consider a square, or an equilateral triangle inscribed in a circle. Do their sides equal to the radius?
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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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In regular hexagon,the length of a diagonal is equal to two times the length of the side, so diagonal is 8 and each side (a) is 4.

Area of hexagon = square root 3*3*a2/2= 24*square root 3.

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Re: If a regular hexagon is inscribed in a circle with a radius of 4, the  [#permalink]

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area of hexagon = 3 * √3/4 * s^2
s=4
answer IMO D; . $$24 \sqrt{3}$$

fozzzy wrote: If a regular hexagon is inscribed in a circle with a radius of 4, the area of the hexagon is

A. $$12 \sqrt{3}$$

B. $$8 \pi$$

C. $$18 \sqrt{2}$$

D. $$24 \sqrt{3}$$

E. 48

Attachment:
screen_shot_2010_12_23_at_3.41.13_pm.png Re: If a regular hexagon is inscribed in a circle with a radius of 4, the   [#permalink] 21 May 2019, 02:13
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