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

Question

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}

Bunuel · Accepted Answer

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.

whiplash2411 · Answer

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 [ 41.88 KiB | Viewed 61786 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 = \(\pi*(2\sqrt{3})^2 = 12\pi\)

rite2deepti · Answer

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 ....

but its not in the answer choices OA is something else ....

Jaisri · Answer

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.

Bunuel · Answer

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.

Bunuel · Answer

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 ....

Kimberly77 · Answer

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

Kimberly77 · Answer

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

Round2Hopeful · Answer

would this be considered a 700 level question?

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.

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

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

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

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