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# A circle is inscribed in an equilateral triangle, Find area

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A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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30 Aug 2014, 05:58
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A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20
B 45
C 60
D 55
E 77
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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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08 Sep 2014, 22:20
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PathFinder007 wrote:
A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20
B 45
C 60
D 55
E 77

PareshGmat's solution is crisp and perfect. Let me just add the explanation for this:

Radius of inscribed circle =$$\frac{\sqrt{3}a}{6}$$

We know that the altitude of the equilateral triangle will be $$\frac{\sqrt{3}a}{2}$$

The altitude will also be the median and will pass through the center of the circle (since it is an equilateral triangle). We know that centroid divides the median in the ratio 2:1. The centroid will be the center of the circle since each median will pass through it due to symmetry. Hence the radius of the circle will be one third of the altitude.

Radius = $$\frac{\sqrt{3}a}{2} * \frac{1}{3} = \frac{\sqrt{3}a}{6}$$
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Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! ##### Most Helpful Community Reply SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1835 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags Updated on: 09 Sep 2014, 19:38 12 PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 Refer diagram below: Attachment: qSvMt.png [ 9.17 KiB | Viewed 39588 times ] Let the dimension of equilateral Triangle = a Area of the triangle$$= \frac{\sqrt{3}}{4} a^2$$ Radius of inscribed circle $$= \frac{\sqrt{3}}{6} a$$ Area of inscribed circle $$= \frac{\pi a^2}{12}$$ Percentage area $$= \frac{\frac{\pi a^2}{12}}{\frac{\sqrt{3} a^2}{4}} * 100 = \frac{\pi}{3\sqrt{3}} * 100 = 62.8%$$ Answer = C _________________ Kindly press "+1 Kudos" to appreciate Originally posted by PareshGmat on 08 Sep 2014, 21:26. Last edited by PareshGmat on 09 Sep 2014, 19:38, edited 1 time in total. ##### General Discussion Intern Joined: 18 Aug 2014 Posts: 11 Location: India Concentration: General Management, Finance GMAT Date: 10-08-2014 GPA: 3.23 WE: Analyst (Retail Banking) A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags Updated on: 09 Sep 2014, 10:27 3 1 PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 Assume this as an equilateral triangle with side as a (Yes I know, this isn't an equilateral triangle hence i asked you to assume, this is the only diagram I got from internet and I am too lazy to draw one myself) Attachment: File comment: Diagram Su55k02_m10 (1).gif [ 9.13 KiB | Viewed 39938 times ] so now area of an equilateral triangle = $$\frac{\sqrt3}{4} * a^2$$ -----1 and area of the triangle is also equal to Area of triangle AOC+Area of AOB + Area of BOC = $$\frac{1}{2} * a * r$$( r is height of individual triangle) ---------2 from equation 1 & 2 above $$\frac{\sqrt3}{4} * a^2 = 3 * \frac{1}{2} * a * r$$ from here we can get the value of a i.e. $$a = 2\sqrt{3} * r$$ ---------3 Now, In the question we need to find out $$\frac{Area of circle inscribed}{Area of the equilateral triangle}$$ which is equal to $$\frac{{\pi * r^2}}{{3* 1/2 * (2 \sqrt3 r)^2}}$$ ------substituting the value of a from equation 3 =$$\frac{\pi}{{3 * \sqrt3}}$$ $$\approx \frac{3.14}{{3 * 1.72}}$$ $$\approx \frac{3.14}{{3 * 1.72}}$$ $$\approx \frac{2}{3} \approx 0.66$$ Notice that we reduced the numerator by $$0.26$$ so our answer is going to be a bit inflated. Looking at the answer choices, C is the closest. (Notice that there's nothing between 60 and 70 in the options so we can be a little imprecise in this case) Hence the solution is C _________________ The buttons on the left are the buttons you are looking for Originally posted by Anamika2014 on 08 Sep 2014, 08:42. Last edited by Anamika2014 on 09 Sep 2014, 10:27, edited 1 time in total. SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1835 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags 08 Sep 2014, 21:46 Anamika2014 wrote: PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 Assume this as an equilateral triangle with side as a (Yes I know, this isn't an equilateral triangle hence i asked you to assume, this is the only diagram I got from internet and I am too lazy to draw one myself) Attachment: Su55k02_m10 (1).gif so now area of an equilateral triangle = $$\frac{\sqrt3}{4} * a^2$$ -----1 and area of the triangle is also equal to Area of triangle AOC+Area of AOB + Area of BOC = $$\frac{1}{2} * a * r$$( r is height of individual triangle) ---------2 from equation 1 & 2 above $$\frac{\sqrt3}{4} * a^2 = 3 * \frac{1}{2} * a * r$$ from here we can get the value of a i.e. $$a = 2\sqrt{3} * r$$ ---------3 Now, In the question we need to find out $$\frac{Area of circle inscribed}{Area of the equilateral triangle}$$ which is equal to $$\frac{{\pi * r^2}}{{3* 1/2 * (2 \sqrt3 r)^2}}$$ ------substituting the value of a from equation 3 =$$\frac{\pi}{{3 * \sqrt3}}$$ \approx \frac{3.14}{{3 * 1.72}} $$\approx \frac{3.4}{{3 * 1.72}} [m]\approx \frac{2}{3} \approx 0.66$$ Notice that we reduced the numerator by $$0.26$$ so our answer is going to be a bit inflated. Looking at the answer choices, C is the closest. (Notice that there's nothing between 60 and 70 in the options so we can be a little imprecise in this case) Hence the solution is C It has to be 3.14 instead of 3.40 _________________ Kindly press "+1 Kudos" to appreciate SVP Joined: 08 Jul 2010 Posts: 2137 Location: India GMAT: INSIGHT WE: Education (Education) Re: A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags 03 Feb 2016, 09:56 1 PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 To answer this question we need (Area of circle/Area of Triangle)*100 If the circle is in Equilateral triangle then (1/3)*Height of equilateral Triangle = Radius of Circle i.e. Radius of Circle = $$(1/3)*(\sqrt{3}/2)*side = Side/2\sqrt{3}$$ (Area of circle/Area of Triangle)*100 = $$(Pi Side^2/12)*100/ \sqrt{3}*Side^2/4 = pi*100/3\sqrt{3} = 60%$$ Answer: Option C _________________ Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html 22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION VP Joined: 07 Dec 2014 Posts: 1066 A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags 10 Mar 2017, 10:07 1 PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 let r=both the radius of the inscribed circle and the base of each of the 6 identical 30-60-90 right triangles subsumed by the equilateral triangle area of circle=⫪r^2 area of equilateral triangle=6*1/2*r*r√3=3r^2√3 ⫪r^2/3r^2√3=⫪/3√3≈.60=60% C Attachments circleintriangle.png [ 30.23 KiB | Viewed 33741 times ] Director Joined: 12 Nov 2016 Posts: 772 Location: United States Schools: Yale '18 GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66 Re: A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags 14 Mar 2017, 23:44 PareshGmat wrote: PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 Refer diagram below: Attachment: qSvMt.png Let the dimension of equilateral Triangle = a Area of the triangle$$= \frac{\sqrt{3}}{4} a^2$$ Radius of inscribed circle $$= \frac{\sqrt{3}}{6} a$$ Area of inscribed circle $$= \frac{\pi a^2}{12}$$ Percentage area $$= \frac{\frac{\pi a^2}{12}}{\frac{\sqrt{3} a^2}{4}} * 100 = \frac{\pi}{3\sqrt{3}} * 100 = 62.8%$$ Answer = C How did you know to apply these formulas? Was there some kind of algebra you did to arrive at root3 divided by 4? I don't understand where you got root 3 divided 4 - I know it works because I plugged it in and tried but don't understand what math I need to do to get there. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8187 Location: Pune, India Re: A circle is inscribed in an equilateral triangle, Find area [#permalink] ### Show Tags 15 Mar 2017, 01:59 2 1 Nunuboy1994 wrote: PareshGmat wrote: PathFinder007 wrote: A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle? A 20 B 45 C 60 D 55 E 77 Refer diagram below: Attachment: qSvMt.png Let the dimension of equilateral Triangle = a Area of the triangle$$= \frac{\sqrt{3}}{4} a^2$$ Radius of inscribed circle $$= \frac{\sqrt{3}}{6} a$$ Area of inscribed circle $$= \frac{\pi a^2}{12}$$ Percentage area $$= \frac{\frac{\pi a^2}{12}}{\frac{\sqrt{3} a^2}{4}} * 100 = \frac{\pi}{3\sqrt{3}} * 100 = 62.8%$$ Answer = C How did you know to apply these formulas? Was there some kind of algebra you did to arrive at root3 divided by 4? I don't understand where you got root 3 divided 4 - I know it works because I plugged it in and tried but don't understand what math I need to do to get there. If the side of an equilateral triangle is s, The altitude of the equilateral triangle $$= \frac{\sqrt{3}}{2} * s$$ (Draw the altitude of an equilateral triangle and then use pythagorean theorem on the right triangle obtained) The area of the equilateral triangle $$= (\frac{1}{2})*base*altitude = (\frac{1}{2})*s*\frac{\sqrt{3}}{2}s = \frac{\sqrt{3}}{4} s^2$$ _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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15 Mar 2017, 18:39
Okay the formulas appear to be more clear and logical now- but my question is how do you divide a number by an imperfect square- for example, in a equilateral triangle with sides of 2, I used the formula root divided by 4 times 2^2 to get the area. The formula needed to solve the problem as stated is area of circle/ area of triangle * 100- so now we have the denominator. Knowing that 1/3 of the height of equilateral triangle is the radius- we can use the Pythagorean Theorem or other formula Karishma stated and arrive at root/ three. The area of the circle is then pi times root/3 squared. We can then simplify and arrive at pi times 1/3 (1/3 is the radius square in simplest form) divided by root 3. On my calculator this equals .60- though how would you make that calculation without a calculator?
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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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15 Mar 2017, 22:50
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Top Contributor
1
Hi,
Since couple of replies already used equilateral triangle formula, to solve the question
Let’s see now how to solve this question, with using the basic area of triangle formula,
If the question asks for percentages or ratios always we can try some number out,
Here say the side of the equilateral triangle is “6”,
We have to find the area of the triangle and area of the circle inscribed,
That is,
(Area of the circle inscribed/Area of the triangle)*100
Area of the circle is (pi)* r^2
Area of the triangle is ½ * base * height

Refer to the diagram,

So, here area of the triangle, ½ * 6 * 3 (root 3) = 9 root 3,
Also remember that, radius of the circle is always 1/3 rd the height of the equilateral triangle (This is because, Medians of the triangle intersects at 1:2 ratio).
So here the radius would be, 1/3(3 root 3) = root 3
So the Area of the circle is, pi * (root 3)^2 = 3 * pi,
We can approximate “pi” value as “3“, because answer choices are wide enough, So then area of the circle approximately is 9.
Now, Area of the triangle, we can approximate as 9 root 3 = 9 * 1.7 = 15.3, So we can further approximate it as 15.
So we can find the percentage now,
(Area of the circle inscribed/Area of the triangle)*100
(9/15)* 100.
This is 60%.
Hope this helps
Attachments

Triangle - inscribed circle.png [ 11.07 KiB | Viewed 33494 times ]

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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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16 Mar 2017, 07:01
VeritasPrepKarishma wrote:
PathFinder007 wrote:
A circle is inscribed in an equilateral triangle, such that the two figures touch at exactly 3 points, one on each side of the triangle. Which of the following is closest to the percent of the area of the triangle that lies within the circle?

A 20
B 45
C 60
D 55
E 77

PareshGmat's solution is crisp and perfect. Let me just add the explanation for this:

Radius of inscribed circle =$$\frac{\sqrt{3}a}{6}$$

We know that the altitude of the equilateral triangle will be $$\frac{\sqrt{3}a}{2}$$

The altitude will also be the median and will pass through the center of the circle (since it is an equilateral triangle). We know that centroid divides the median in the ratio 2:1. The centroid will be the center of the circle since each median will pass through it due to symmetry. Hence the radius of the circle will be one third of the altitude.

Radius = $$\frac{\sqrt{3}a}{2} * \frac{1}{3} = \frac{\sqrt{3}a}{6}$$
PareshGmat's solution is comprehensive ..Could you please give some similar problems Like it...I think this is harder than OG books questions.
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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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16 Mar 2017, 08:09
Nunuboy1994 wrote:
Okay the formulas appear to be more clear and logical now- but my question is how do you divide a number by an imperfect square- for example, in a equilateral triangle with sides of 2, I used the formula root divided by 4 times 2^2 to get the area. The formula needed to solve the problem as stated is area of circle/ area of triangle * 100- so now we have the denominator. Knowing that 1/3 of the height of equilateral triangle is the radius- we can use the Pythagorean Theorem or other formula Karishma stated and arrive at root/ three. The area of the circle is then pi times root/3 squared. We can then simplify and arrive at pi times 1/3 (1/3 is the radius square in simplest form) divided by root 3. On my calculator this equals .60- though how would you make that calculation without a calculator?

You are asked for the closest value, so approximate.
sqrt(2) = 1.4
sqrt(3) = 1.7

If needed, these values are likely to be given in the question anyway.

Though most likely, options in actual GMAT questions will retain the irrational numbers. So you would probably see options in terms of sqrt(3).
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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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17 Mar 2017, 03:45
let the side of the triangle be a
area = √3 a^2/4
median length = √3 a/2
radius = 1/3 *√3 a/ 2 = a/ 2√3

area of circle = pi *a/2√3 * a/2√3 = pi * a^2/ 12

therefore percentage = (pi a^2/12)/ √3 a^2/4

around 60%
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Re: A circle is inscribed in an equilateral triangle, Find area  [#permalink]

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12 Apr 2018, 08:55
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