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What is the perimeter of an equilateral triangle inscribed

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What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post Updated on: 09 Aug 2012, 15:05
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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

Originally posted by arthuro69 on 09 Aug 2012, 13:06.
Last edited by Bunuel on 09 Aug 2012, 15:05, edited 1 time in total.
Renamed the topic and edited the question.
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post 09 Aug 2012, 16:29
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arthuro69 wrote:
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


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:
http://magoosh.com/gmat/2012/the-gmats- ... triangles/

Let me know if there are any further questions.

Mike :-)
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post 04 Sep 2012, 03:29
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).
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post 04 Sep 2012, 03:32
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post 06 Jul 2014, 11:30
Let x be the side of triangle.

Using cosine rule:
(x/2)/R = cos 30
=> x = 2Rcos 30 = 4√3
=> Perimeter = 3X = 12√3
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post 19 Apr 2018, 14:23
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arthuro69 wrote:
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. \(24\)


So, here's what the diagram looks like.
Image

If we draw lines from the center to each vertex, we get the following:
Image

Since the radii have length 4, we can add that here:
Image

Now we'll draw a line from the center that is PERPENDICULAR to one side of the tirangle.
Image
We now have a SPECIAL 30-60-90 right triangle.

Here's the base version of this SPECIAL TRIANGLE
Image

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
Image

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
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Re: What is the perimeter of an equilateral triangle inscribed  [#permalink]

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New post 16 Jul 2019, 10:45
arthuro69 wrote:
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


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.
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Re: What is the perimeter of an equilateral triangle inscribed   [#permalink] 16 Jul 2019, 10:45
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