If a regular hexagon is inscribed in a circle with a radius of 4, the

Question

https://gmatclub.com/forum/download/file.php?id=20225 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 screen_shot_2010_12_23_at_3.41.13_pm.png

BangOn · Accepted Answer

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

Zarrolou · Answer

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}

gultrage · Answer

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

Bunuel · Answer

Questions about hexagons: 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.

koolgmat · Answer

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

lchen · Answer

1) Each of the hexagon's angles 120 degrees : formula if you don't know it is   = (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
Answer is  D

fozzzy · Answer

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

gracie · Answer

the hexagon comprises 6 identical equilateral triangles
area of 1 equilateral triangle=√3/4*4^2=4√3
6*4√3=24√3
D

JeffTargetTestPrep · Answer

We may recall that a regular hexagon can be divided into 6 equilateral triangles, and thus the area of a regular hexagon is:
 
6 , 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 
 
Area = 6 
 
Area = 6(4√3)
 
Area = 24√3
 
Answer: D

JesusWalks · Answer

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

Bunuel · Answer

No. Consider a square, or an equilateral triangle inscribed in a circle. Do their sides equal to the radius?

Raxit85 · Answer

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.

Posted from my mobile device

Archit3110 · Answer

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

DevanshiMehrotra · Answer

For a hexagon inscribed in a circle, the radius of the circle is equal to the side of the hexagon. Therefore, in this situation, side of hexagon is 4. Formula for area of hexagon is ((3*square-root 3)/2)*a^2. Put a=4. 
area of hexagon= (3*square-root 3*4^2)/ 2= 24 square-root 3

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If a regular hexagon is inscribed in a circle with a radius of 4, the

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If a regular hexagon is inscribed in a circle with a radius of 4, the

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards