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An equilateral triangle that has an area of 9*root(3)

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Hussain15
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An equilateral triangle that has an area of 9*3^(1/2) [#permalink] New post   07 Jul 2010, 10:45
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An equilateral triangle that has an area of \(9\sqrt{3}\) is inscribed in a circle. What is the area of the circle?

A. \(6\pi\)

B. \(9\pi\)

C. \(12\pi\)

D. \(9\pi \sqrt{3}\)

E. \(18\pi \sqrt{3}\)
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Bunuel
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An equilateral triangle that has an area of 9*root(3) [#permalink] New post   25 Nov 2010, 02:30
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rite2deepti wrote:
An equilateral triangle that has an area of \(9\sqrt{3}\) is inscribed in a circle. What is the area of the circle?

A. \(6\pi\)
B. \(9\pi\)
C. \(12\pi\)
D. \(9\pi \sqrt{3}\)
E. \(18\pi \sqrt{3}\)

This is how I solved it area of equilateral triangle =Square root 3/4*a^2=9 square root 3
we get a =6
know to calculate radius of an equlateral triangle in an inscribed circle we can use formulae

r=a*square root 3/6
with this I get r=square root 3 and then area =3 pi
:( but its not in the answer choices
OA is something else ....


I'd recommend to know the ways some basic formulas can be derived in geometry rather than memorizing them.

Anyway, \(area_{equilateral}=a^2*\frac{\sqrt{3}}{4}\), where \(a\) is the length of a side --> as given that \(area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=9\sqrt{3}\) then \(a=6\);

Now, given that this triangle is inscribed in circle (not circle is inscribed in triangle). The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}=2\sqrt{3}\) (you used the formula for the radius of the inscribed circle \(r=a*\frac{\sqrt{3}}{6}\)) --> \(area_{circle}=\pi{R^2}=12\pi\).

Answer: C.

Check Triangles chapter of Math Book for more: math-triangles-87197.html

Hope it helps.
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Re: An equilateral triangle that has an area of 9*3^(1/2) [#permalink] New post   07 Jul 2010, 11:36
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Given that the area of the equilateral triangle is \(9\sqrt{3}\).

Assuming the side of the triangle is a, we get:

\(\frac{1}{2}*\frac{\sqrt{3}}{2}*a = 9\sqrt{3}\)

Solving this we get \(a = 6\)

Now, in the figure, consider the shaded triangle. Clearly we can see that the area of this triangle is a third of the area of the whole triangle.
Attachment:
EqTriangle.JPG
EqTriangle.JPG [ 41.88 KiB | Viewed 58413 times ]


So we have \(\frac{1}{2}*h*\frac{a}{2} = \frac{1}{3}*9\sqrt{3}\) so from here we get \(h=\sqrt{3}\)

This means that the radius, as we can see from the picture = \(3\sqrt{3}-\sqrt{3}=2\sqrt{3}\)

Hence area of circle = \(\pi*(2\sqrt{3})^2 = 12\pi\)
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An equilateral triangle that has an area of 9*root(3) [#permalink] New post   25 Nov 2010, 02:11
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An equilateral triangle that has an area of \(9\sqrt{3}\) is inscribed in a circle. What is the area of the circle?

A. \(6\pi\)
B. \(9\pi\)
C. \(12\pi\)
D. \(9\pi \sqrt{3}\)
E. \(18\pi \sqrt{3}\)

This is how I solved it area of equilateral triangle =Square root 3/4*a^2=9 square root 3
we get a =6
know to calculate radius of an equlateral triangle in an inscribed circle we can use formulae

r=a*square root 3/6
with this I get r=square root 3 and then area =3 pi
:( but its not in the answer choices
OA is something else ....
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Re: An equilateral triangle that has an area of 9*root(3) [#permalink] New post   28 Jul 2013, 23:37
rite2deepti wrote:
An equilateral triangle that has an area of \(9\sqrt{3}\) is inscribed in a circle.


I am trying to relearn about the term "inscribed". Please help:
Does "inscribed" mean all the edges of the triangle are just touching the circle or can it mean that the trianlge lies completely inside the circle (meaning the edges need not touch the circle and can range from very small size to size that exactly fits in the circle).

In some questions I have seen that it is necessary to assume that inscribed means the triangle completely lies inside (and not necessarily have the edges touch the circle) while for this question is means a perfectly inscribed triangle.
Same doubt applies for circumscribed as well.
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Bunuel
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Re: An equilateral triangle that has an area of 9*root(3) [#permalink] New post   28 Jul 2013, 23:44
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Jaisri wrote:
rite2deepti wrote:
An equilateral triangle that has an area of \(9\sqrt{3}\) is inscribed in a circle.


I am trying to relearn about the term "inscribed". Please help:
Does "inscribed" mean all the edges of the triangle are just touching the circle or can it mean that the trianlge lies completely inside the circle (meaning the edges need not touch the circle and can range from very small size to size that exactly fits in the circle).

In some questions I have seen that it is necessary to assume that inscribed means the triangle completely lies inside (and not necessarily have the edges touch the circle) while for this question is means a perfectly inscribed triangle.
Same doubt applies for circumscribed as well.


A triangle inscribed in a circle does NOT mean that it is simply "inside" the circle, it means that the triangle's vertices are on the circumference of the circle.
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An equilateral triangle that has an area of 9*root(3) [#permalink] New post   21 Nov 2013, 01:49
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An equilateral triangle that has an area of \(9\sqrt{3}\) is inscribed in a circle. What is the area of the circle?

A. \(6\pi\)
B. \(9\pi\)
C. \(12\pi\)
D. \(9\pi \sqrt{3}\)
E. \(18\pi \sqrt{3}\)

\(area_{equilateral}=a^2*\frac{\sqrt{3}}{4}\), where \(a\) is the length of a side --> as given that \(area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=9\sqrt{3}\) then \(a=6\);

We are given that this triangle is inscribed in circle. The radius of the circumscribed circle is \(R=a*\frac{\sqrt{3}}{3}=2\sqrt{3}\) (the radius of the inscribed circle \(r=a*\frac{\sqrt{3}}{6}\)) --> \(area_{circle}=\pi{R^2}=12\pi\).

Answer: C.

Check Triangles chapter of Math Book for more: https://gmatclub.com/forum/math-triangles-87197.html
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Re: An equilateral triangle that has an area of 9*root(3) [#permalink] New post   11 Apr 2022, 14:06
Hi whiplash2411, not sure below calculation of a/2 and 1/3? Could you help explain? Thanks

1/2∗h∗a/2=1/3∗9√3
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An equilateral triangle that has an area of 9*root(3) [#permalink] New post   Updated on: 22 Apr 2022, 13:50
Hi brunel, a bit lost in this part r=a∗√3/6 > r=√3 then. Does it mean I will need to *2 in order to get 2√3? Thanks for your time. Thanks

Originally posted by Kimberly77 on 11 Apr 2022, 14:07.
Last edited by Kimberly77 on 22 Apr 2022, 13:50, edited 1 time in total.
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Re: An equilateral triangle that has an area of 9*root(3) [#permalink] New post   13 Apr 2022, 18:45
would this be considered a 700 level question?
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Re: An equilateral triangle that has an area of 9*root(3) [#permalink] New post   08 Jul 2023, 18:02
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Re: An equilateral triangle that has an area of 9*root(3) [#permalink]
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