A circle O with a radius of 4 is inscribed by a regular hexagon. What

Question

A circle O with a radius of 4 is inscribed by a regular hexagon. What is the length of the perimeter of the regular hexagon? 6.8PS.png A. 16 B. 14√2 C. 16√3 D. 18√5 E. 20√6

yashikaaggarwal · Answer

A regular hexagon consist of 6 equilateral triangle.
Since the circle is inscribed in the hexagon
All sides of hexagon is tangent to the circle.
FE is Tangent to the circle. Hence H is perpendicular on Line segment FE.
Triangle OFE is an equilateral triangle.
H is a perpendicular.
Triangle OHE is a 30-60-90 right angle triangle.
The sides of 30-60-90 triangle are in ratio of p:√3p:2p
Here 4 is corresponding side to 60° hence it is equal to √3P
√3P = 4
P = 4/√3
ALSO, P = X/2 (line corresponding to 30°)
4/√3 = X/2
8/√3 = X
Side of hexagon is 8√3
Perimeter of Hexagon = 6*8/√3
=> 2*3*8/√3
=> 16√3

Answer is C

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MathRevolution · Answer

=>

Since  ∠EOF  is  60° , we have  ∠EOH = 30°.

Then triangle  EOH  is a right triangle with  ∠O = 30°, ∠E = 60° , and  ∠H = 90°.

When we put  OE = x,  we have  HE = \frac{x}{2} = \frac{4}{√3}  or  x = \frac{8}{√3} = \frac{8√3}{3}.

Thus, the perimeter of the regular hexagon is  6·(\frac{8√3}{3}) = 16√3.

Therefore, C is the answer.
Answer: C

ThatDudeKnows · Answer

I adore questions like this. Doing the "real" math takes longer and is FAR more likely to result in a silly mistake than just ballparking and getting the right answer in less than 20 seconds. Why would we do the "real" math?

OH is 4. How long is OE? I don't know, a little longer than 4. Let's just call it 4.5 and see if process of elimination gets us to the finish line.
OE is 4.5, so the perimeter of the hexagon is 6*4.5 = 27.
Let's look at the answer choices.

A.  16 
Wrong.
 
B.  14√2 
 √2  is roughly 1.4, so this is roughly 19-20. Wrong.

C.  16√3  
 √3  is roughly 1.7, so this is roughly 27. Keep it.

D.  18√5 
 √5  is greater than 2, so this is greater than 36. Wrong.

E.  20√6 
This is even larger than D. Wrong.

Answer choice C.

No bonus points for doing the "real" math!

ThatDudeKnowsBallparking

IanStewart · Answer

You could even pick C here without estimating -- A is far too small, and the other answers all contain a square root. A regular hexagon is made up of six equilateral triangles, so when you divide it up, you'll get 30-60-90 triangles, and the only root you'd ever see is √3, never √2, √5 or √6, so only C could conceivably be right here.

But this is a prep company question, not a real GMAT question, and if we're going to evaluate how well a 'look at the answer choices' strategy works, we need to evaluate that strategy only using official problems, because prep companies often don't design answer choices the way the real test does. I'd never expect only one answer to include √3 on an official question like this, so I'd never expect the strategy I just used to work on a real GMAT question, nor would I ever expect an estimate to get the answer to a question specifically like this one (thought it would often get you to 50-50, and is a good fallback strategy). Estimation in geometry can get you the right answer though on lower level geometry questions.

ThatDudeKnows · Answer

I think it's a common misconception that ballparking isn't effective on official questions or that it isn't effective on more difficult official questions. I'll do a thorough breakdown on non-geometry questions at some point, but since this happens to be a geometry question, check THIS out!!

I can't find my breakdown for the main OG, but here's what I've got from the 2022 OG Quant Review. Getting good at estimation, logic, and using the answer choices is a HUGE leg up on geometry questions of ALL difficulty levels, and just might be more useful on "Hard" questions than it is on easier ones.

GMAC has 77 total questions under the Medium Difficulty section (their difficulty rating, not mine). By my count 17 of them are (non-coordinate) geometry questions.  10 of the 17 can be solved with an absolute minimum of "real" geometry, and ballparking alone works on several. 
#89:  https://gmatclub.com/forum/if-a-rectangle-of-area-24-can-be-partitioned-into-exactly-3-nonoverlap-294398.html#p3028424 
#90:  https://gmatclub.com/forum/in-the-figure-above-the-area-of-the-parallelogram-is-294399.html#p3015027 
#92:  https://gmatclub.com/forum/in-the-figure-above-if-triangles-abc-acd-and-ade-are-isosceles-righ-294401.html#p3015032 
#93:  https://gmatclub.com/forum/each-of-27-white-1-centimeter-cubes-will-have-exactly-one-face-painted-294403.html#p2282977 
#96:  https://gmatclub.com/forum/which-of-the-following-gives-all-possible-values-of-x-in-the-figure-ab-294406.html#p2266389 
#102:  https://gmatclub.com/forum/the-figure-above-represents-an-antenna-tower-with-two-guy-wires-that-e-294413.html#p2267425 
#104:  https://gmatclub.com/forum/the-figure-above-shows-a-drop-leaf-table-with-all-four-leaves-down-th-294415.html#p3015035 
#105:  https://gmatclub.com/forum/if-the-diameter-of-a-circular-skating-rink-is-60-meters-the-area-of-294416.html#p2282983 
#129:  https://gmatclub.com/forum/an-open-box-in-the-shape-of-a-cube-measuring-50-centimeters-on-each-si-272045.html#p3015041 
#151:  https://gmatclub.com/forum/in-the-figure-shown-a-square-grid-is-superimposed-on-the-map-of-a-par-272219.html#p2105546

GMAC has 54 total questions under the Hard Difficulty section (their difficulty rating, not mine). By my count 9 of them are (non-coordinate) geometry questions.  7 of the 9 can be solved with an absolute minimum of "real" geometry, and ballparking alone works on several. 
#160:  https://gmatclub.com/forum/the-figure-shown-above-consists-of-three-identical-circles-that-are-ta-168576.html#p3015057 
#167:  https://gmatclub.com/forum/a-window-is-in-the-shape-of-a-regular-hexagon-with-each-side-of-length-210594.html#p1942957 
#168:  https://gmatclub.com/forum/in-the-figure-shown-pqrstu-is-a-regular-polygon-with-sides-of-lenght-x-220821-20.html#p3015079 
#174:  https://gmatclub.com/forum/in-pentagon-pqrst-pq-3-qr-2-rs-4-and-st-5-which-of-the-168634.html#p3015085 
#177:  https://gmatclub.com/forum/when-the-figure-above-is-cut-along-the-solid-lines-folded-along-the-201429.html#p1578865 
#180:  https://gmatclub.com/forum/the-points-r-t-and-u-lie-on-a-circle-that-has-radius-4-if-the-length-168676.html#p3015087 
#206:  https://gmatclub.com/forum/a-square-wooden-plaque-has-a-square-brass-inlay-in-the-center-leaving-89215-20.html#p3028429

Trust me: get good at ballparking, logic, and using the answer choices!!

IanStewart · Answer

If you reread what I wrote, I think you'll recognize that you're mischaracterizing what I said. I've studied the effectiveness of estimation (and of every other 'strategy' recommended in prep books) using large pools of official questions. And it turns out estimation (combined with a bit of logic) is, by far, the most successful guessing strategy on the GMAT. In fact, for PS, it's really the only effective guessing strategy. I've taken repeated official practice tests where I rely only on logic and estimation (and a random number generator when I need to guess) for PS, and on a guessing strategy for DS I won't describe here, and without solving anything I can score in the Q40-Q44 range consistently. I'm defining 'estimation' a bit loosely, so I'm including basic conceptual reasoning (e.g. in a weighted average situation, I'm using the fact that we know the overall average is closer to the larger group's average). That's applying those conceptual principles and estimates perfectly, so it's not the score someone should expect applying those principles imperfectly, but it's still very effective.

The problem is, I can't get beyond that level. And there's a reason for that: if an estimation strategy works on, say, this hexagon question in this thread (and the hexagon is approximately the circle, so estimation clearly will get you very close), every high level test taker will see that, even the high level test takers who can't see how to 'do the math'. So every high-level test taker will get the question right, one way or another. And that means the question cannot be a high-level question, because that's how they determine question difficulty: they administer questions as experimental questions, and see how well test takers at each ability level answer them. If every high-level test taker gets a certain question right, it is essentially by definition not a high-level question. So these strategies do start to fail the harder questions get.

To me the best practice is this: if solving seems annoying or time-consuming, glance at answer choices. If they're far apart, estimate liberally. If they're close together, you can't be sure, after estimating, that you'll be able to identify the right answer, so use it as a fallback only, if you can't find a better way to answer. And get good at math (by which I mostly mean: get good at conceptual reasoning), because that's what the hardest GMAT questions are really testing.

Anyway, I just wanted anyone reading this discussion to have two points of view, from two people who have thought about these questions a lot.

ThatDudeKnows · Answer

Not to be pedantic, but I'm certainly not suggesting relying "only on logic and estimation." In fact, I noted that logic, estimation, and using the answer choices work on 10 out of 17 medium official questions and 7 out of 9 hard official questions. The implication is that you'll need to do real math on the others, but 16 out of 26 is  61% of   medium and hard   geometry questions , so it's a heck of a lot better than "never," it certainly applies to more than just "lower level geometry questions," and it is without a doubt the single most useful thing you can learn in geometry (more valuable than any content-based studying). When you CAN use those tools, they allow you to answer questions more quickly (so you have more time for questions that do require real math), cause far fewer silly mistakes (which directly improves your score), and avoid the stress and brain damage that happens when you're crunching numbers and find yourself stuck (which helps you stay on track mentally through the rest of exam, indirectly improving your score).

IanStewart · Answer

Again, I'd ask you not to mischaracterize what I've written. For the benefit of anyone reading this far, I was discussing how successful an estimation and logic based strategy can be. I didn't say anything about you at all, let alone about what strategies you might recommend.

You also seem to think I'm opposed to estimation, even though I just said the opposite, more or less -- it can be a very useful technique, though it has its limitations. I've been posting here for more than ten years, and if you read even a few of my solutions to Quant problems on this forum, I think you'll find that when estimation is the best technique, I use it, and when it isn't, I don't.

I'm all for good faith conversations about what strategies work and what don't on the GMAT, but I don't really have time to correct people who claim I've written things I haven't, so perhaps it's a good idea to end the conversation here, or if you want to reply, I'd be grateful if you could try not to paraphrase me when you do.

bumpbot · Answer

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A circle O with a radius of 4 is inscribed by a regular hexagon. What

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A circle O with a radius of 4 is inscribed by a regular hexagon. What

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