What is the perimeter of an equilateral triangle inscribed

Question

What is the perimeter of an equilateral triangle inscribed in a circle of radius 4 ?

A.  6\sqrt{2} 
B.  6\sqrt{3} 
C.  12\sqrt{2} 
D.  12\sqrt{3} 
E.  24

source: readyforgmat.com

BrentGMATPrepNow · Accepted Answer

So, here's what the diagram looks like. 
 https://i.imgur.com/Mgsgd9t.png

If we draw lines from the center to each vertex, we get the following:
 https://i.imgur.com/dHG8OFM.png

Since the radii have length 4, we can add that here:
 https://i.imgur.com/BHpT2Gv.png

Now we'll draw a line from the center that is PERPENDICULAR to one side of the tirangle.
 https://i.imgur.com/jqbPaMa.png 
We now have a SPECIAL 30-60-90 right triangle.

Here's the base version of this SPECIAL TRIANGLE
 https://i.imgur.com/bMvROpa.png

We can see that the each 30-60-90 triangle in the diagram is TWICE as big as the base version. So, each side opposite the 60º angle must have length 2√3
 https://i.imgur.com/fX01R1P.png

This means ONE side of the equilateral triangle has length 4√3, so the PERIMETER = 4√3 + 4√3 + 4√3 = 12√3

Answer: C

Cheers,
Brent

mikemcgarry · Answer

Hi, there. I'm happy to help. :-)

The full solution to the problem is in the attached pdf. If the details of the 30-60-90 triangle are not familiar to you, I recommend brushing up with this post: https://magoosh.com/gmat/2012/the-gmats- ... triangles/ Let me know if there are any further questions. Mike :-)

venuvm · Answer

The radius of circum circle of an equilateral triangle = a/sqrt(3). a is the side of triangle.
Here: a/sqrt(3) = 4
a = 4*sqrt(3).
perimeter 3a = 3*4*sqrt(3) = 12sqrt(3).

Bunuel · Answer

All you need to know about triangles for the GMAT: math-triangles-87197.html Hope it helps.

anujbansal · Answer

Let x be the side of triangle.

Using cosine rule: 
(x/2)/R = cos 30
=> x = 2Rcos 30 = 4√3 
=> Perimeter = 3X = 12√3

thelion10 · Answer

way simpler to simply know a formula :

Whenever a Circle circumscribes an equilateral triangle, the The radius of the circle can be defined by : Radius =  \frac{Side Of The Triangle}{\sqrt{3}}  .

that way, if we solve the problem : 4 =  \frac{s}{\sqrt{3}}  . Hence, side =  4\sqrt{3} .

Now, the perimeter = 3 x  4\sqrt{3}  = 12 \sqrt{3}  . 
The answer is the option D.

adityasuresh · Answer

For equilateral triangle the radius of the circumscribed circle R = Side *  Root 3/3

What is the perimeter of an equilateral triangle inscribed

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What is the perimeter of an equilateral triangle inscribed

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