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If each side of parallelogram P has length 1, what is the area of P ?

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woohoo921
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Re: If each side of parallelogram P has length 1, what is the area of P ? [#permalink] New post   21 Aug 2021, 13:42
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EMPOWERgmatRichC wrote:
Hi All,

We're told that each side of parallelogram P has length 1. We're asked for the area of P. This is a great 'concept question', meaning that if you know the concepts involved, then you don't actually have to do much math to get the correct answer. Here, since we know all 4 sides of the parallelogram, if we have ANY of the 4 angles, then can determine the area.

(1) One angle of P measures 45 degrees.

A parallelogram is 360 degrees and 'opposite' angles are equal. With the information in Fact 1, we know that the 4 angles are 45/135/45/135. Combined with the side lengths (which we know are all 1s), we can determine the exact area of this shape.
Fact 1 is SUFFICIENT

(2) The altitude of P is √2/2

The information in Fact 2 requires a bit more work, but we can now draw 2 RIGHT triangles "inside" the parallelogram. Each of those right triangles would have a hypotenuse of 1 (since the side lengths are 1s) and a 'height' of √2/2. We could determine the exact value of the 3rd side and the two angles (it ends up being a 45/45/90 right triangle), so we can then determine the area of the overall shape.
Fact 2 is SUFFICIENT

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You are always so helpful... I have a clarifying question based on reading the Official Guide's answer. To actually calculate the area is it just (1)(1/root(2))? I see the answer as (root(2))/2, if my understanding is correct, why did the Official Guide feel the need to multiple the 1/root(2) times root(2) in the numerator and denominator... is it because you never want an answer with a radical in the bottom? Thank you so much in advance :)
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Re: If each side of parallelogram P has length 1, what is the area of P ? [#permalink] New post   21 Aug 2021, 16:59
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Hi woohoo921,

Yes - to both of your questions. The area of a parallelogram is (Base)(Height) - and by the "Height", we really mean the "Altitude." As a rule, it's generally frowned upon to have a radical in the denominator of a fraction - so 'canceling it out' (by multiplying by another radical) is often done. I've seen some rare cases in which that extra step was not included, but you'll be far more likely to need to do that extra step on Test Day than not.

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Re: If each side of parallelogram P has length 1, what is the area of P ? [#permalink] New post   05 Feb 2022, 19:49
question on this. If item 1 had an angle other than 30, 45, or 60 (those being the angles that result in 30-60-90 or 45-45-90 triangles), would 1 alone have still been sufficient? For instance, say 1 is 65 degrees, we wouldn't have DS on this piece right?

a^2 + b^2 = 1 ^ 2

We wouldn't have enough to solve A or B in parth. theorem right?
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Re: If each side of parallelogram P has length 1, what is the area of P ? [#permalink] New post   07 Feb 2022, 12:35
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gmatmba12 wrote:
question on this. If item 1 had an angle other than 30, 45, or 60 (those being the angles that result in 30-60-90 or 45-45-90 triangles), would 1 alone have still been sufficient? For instance, say 1 is 65 degrees, we wouldn't have DS on this piece right?

a^2 + b^2 = 1 ^ 2

We wouldn't have enough to solve A or B in parth. theorem right?


Hi gmatmba12,

Since we know that all 4 sides of the parallelogram are equal to 1, if we have ANY of the 4 angles, then we can determine the area of the parallelogram. With the 'magic' Pythagorean triangles (re: 45/45/90 and 30/60/90), the math work is far easier - but ANY angle would be enough (although we'd end up using Trigonometry rules with any non-magic angles, such as 65).

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Contact Rich at: Rich.C@empowergmat.com
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Re: If each side of parallelogram P has length 1, what is the area of P ? [#permalink] New post   24 Feb 2024, 06:10
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Re: If each side of parallelogram P has length 1, what is the area of P ? [#permalink]
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