Mini-Me, You Complete Me: Similar Shapes On The GMAT

Avi Gutman —  September 26, 2012 — 11 Comments

gmat similar shapes

Try to solve the following question, and time yourself:

If the volume of a big cube is 64 times that of a small cube, how many times bigger is the surface area of the big cube than that of the small cube?

If you cannot answer the above (classic GMAT) question in under 20 seconds, continue reading and you will learn a concept that will be super useful in your quest to crush the GMAT!

I was watching Austin Powers the other day and it suddenly hit me: Dr. Evil and Mini-Me are similar shapes! You know, like similar triangles, where the proportion between any two matching sides is always maintained “ if Mini-Me’s fingers are exactly half the length of Dr. Evil’s fingers, then Mini-Me’s eyes, ears, nose, and feet must also be exactly half their counterparts in Dr. Evil’s body. It got me thinking “ what other kinds of similar shapes could be out there? I will investigate that thought further in the second half of this post, but first let’s see why that might be useful

We know triangles are similar whenever they have the same three angles. If the base of the bigger triangle is exactly twice that of the smaller triangle, then each side in the bigger triangle will also be twice as big as its matching side in the smaller triangle.

 

gmat similar triangles

Intuitively, if every side in the bigger triangle is exactly twice as big as its matching side in the smaller triangle, then the sum of those bigger triangle sides will also be twice as big as the sum of the matching smaller triangle sides. In other words, the bigger perimeter will be twice as big as the smaller perimeter:

 2a + 2b + 2c = 2 Ã— (a + b + c)

I can also imagine that the bigger triangle’s heights will be twice as big as their matching heights in the smaller triangle (they are). So, we may conclude that the proportion between the similar triangles will remain the same for any matching lengths (1-dimensional) we can think of!

Would this same proportion extend also to the areas (2-dimensional) of the triangles? The area of a triangle is half the product of a base and its height (A = \frac{b*h}{2}); if the bases and heights of the bigger triangle are n times bigger than those of the smaller triangle, then the formula would produce a result that is n^2 bigger for the bigger triangle:

Area of big triangle = \frac{(nb)*(nh)}{2} = n^2 * \frac{b*h}{2}

Area of small triangle = \frac{b*h}{2}

So for any 1-dimensional ratio, all we have to do is square it and we’ll get the 2-dimensional ratio: if the ratio of the lengths is 2:1, the 2-dimensional ratio would be 2^2:1^2:, or 4:1. This result is not that surprising if we think about how we measure different dimensions in real life: length in feet and area in square feet. How do we measure volume (3-dimensional) in real life? You got it “ cubic feet! I won’t show the proof here, but using the same intuition we might conclude (correctly) that the ratio of volumes of similar shapes will be the ratio of lengths cubed.

What about circles? They are much more simple than triangles because they are determined by only one parameter “ if you give me the radius I really don’t need anything else. Does this mean all circles in the universe are similar? Absolutely “ if the radius of a big circle is twice as big as that of a small circle, then the diameter and circumference of the big circle will also be exactly twice as big as those of the small circle (try plugging numbers and see for yourself). And squares? They too are determined by only one parameter “ if you tell me the length of a side of a square I know exactly what the square will look like So squares are automatically similar, just like circles. Other shapes can be similar as well, under the right circumstances “ let us know in the comments if you can think of any!

Now, let’s see if we can solve that classic GMAT question from the beginning of this post Here it is again, for your convenience:

If the volume of a big cube is 64 times that of a small cube, how many times bigger is the surface area of the big cube than that of the small cube?

Cubes are 3-dimensional squares, so they are automatically similar (defined by one parameter). We are told the volume (3-dimensional) ratio between the cubes is 64:1, therefore the 1-dimensional ratio must be the cube root of that ratio, or 4:1 (you should have 4^3=64 memorized). We were asked to find the ratio of the surface areas (2-dimensional). Now that we know the 1-dimensional ratio is 4:1, finding the 2-dimensional ratio is a piece of cake! 4^2:1^2 = 16:1 “ the answer to the question is 16.

In the several years that I’ve been teaching, I have seen countless GMAT questions that are very difficult (or time consuming) to solve conventionally, and very easy to solve using similar shape ratios (and converting them among the 3 dimensions). My conclusion is that the GMAT writers implicitly test our understanding of this concept, which is what prompted me to write about it.

Takeaways:

  1. Similar shapes share a very important characteristic: they maintain a constant ratio between any matching lengths, and that ratio is squared for matching areas and cubed for matching volumes.
  2. Similar shapes come in many different forms, so be on the lookout: all squares (and cubes) are similar, all circles (and spheres) are similar, all triangles that have the same three angles are similar, and (OK I’ll give you guys one more) all rectangles that have the same length : width ratio are similar.
  3. Whenever you notice that a question involves similar shapes on the GMAT, try to use this concept in your solution. Let us know in the comments if you come across any in the official guides!

Avi Gutman

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Avi Gutman earned his MBA at NYU’s Stern School of Business. While at Stern, he took on the position of teaching fellow and taught his classmates several courses including Operations, Foundations of Finance, and New Venture Financing. Avi has been teaching test prep since 2004 and has worked for several different companies before joining Manhattan GMAT.

11 responses to Mini-Me, You Complete Me: Similar Shapes On The GMAT

  1. Mr. Gautam:

    Could you also share your GMAT score and the number of attempts you had to take to achieve that score?

    Thank you,
    Alex

  2. Hi Curious Alex,
    I was teaching for the (Israeli) SATs when I took my first GMAT, so I was already well prepared and scored 99% on my first attempt. I have continued to score 99% every time since.

  3. Hi Avi,
    If say there are two cubes.2nd cube’s volume is twice the surface area of 1st cube,what is the volume of 2nd cube in terms of V?
    I tried using the following: The surface area ratio would be 2:1, so volume will be in the ratio 8:1,but i am not sure how to proceed further.
    Thanks,
    Shreeraj

  4. Oh ok,i think i found my mistake.Since the surface area is in the ratio 2:1 (2 dimensional),the 1 dimensional ratio would be root2:1 and then since we want the volume(3 dimensional),cube it to get 2root2:1 which is the correct answer.Am i right here?

    Thanks,
    Shreeraj

  5. That is exactly right, well done Shreeraj!

  6. Dear Sheeraj,
    I think there’s a problem in your original question (‘m sure it may be typing mistake). Pls see below.

    your original question goes like this “If say there are two cubes.2nd cube’s volume is twice the surface area of 1st cube,what is the volume of 2nd cube in terms of V?”
    But I feel the correct question should be like this
    “If say there are two cubes.2nd cube’s ‘surface area’ (not volume) is twice the surface area of 1st cube,what is the volume of 2nd cube in terms of V?

    I highlighted because a wrong question will lead to a wrong answer & wrong understanding.

    Avi, you may pls correct me.

    Thanks
    Niladri

  7. Hi Niladri,
    You are absolutely correct.
    I had assumed Shreeraj had a typo but I should have called it out because as you mention in your comment, it could lead to misunderstandings.
    Thank you for pointing it out!

  8. Oops,my bad,thanks for correcting Niladri!

  9. hey avi, i didnt got ans to the question made by shreeraj can u please explain me?
    thank you
    kesha

  10. Hey Kesha,
    Shreeraj’s question was (in my own words):
    A large cube’s surface area is two times greater than that of a small cube; how many times greater is the volume of the large cube than that of the small cube?
    Since all cubes in the world are similar (see logic in post), and since we are ‘told’ that the 2-dimensional ratio between the cubes is 2:1 (area is 2-dimensional), we can infer that the 1-dimensional ratio is the square root of that ratio (so root2:1) and in order to get the 3-dimensional ratio (volume is 3-dimensional) we must cube the 1-dimensional ratio (see logic in post) and we get the answer: (root2)^3 : 1^3 = 2*root2 : 1
    Let me know if this clarifies!
    Cheers,
    Avi

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