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555-605 (Medium)|   Geometry|      
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Asifpirlo
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In a particular 4-sided pyramid, each side of the square bas Updated on: 16 Aug 2013, 14:53
Originally posted by Asifpirlo on 15 Aug 2013, 10:28.
Last edited by Asifpirlo on 16 Aug 2013, 14:53, edited 1 time in total.
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In a particular 4-sided pyramid, each side of the square base is 50 feet in length. If the apex of the pyramid is 60 feet from the ground, what is the total surface area of the pyramid, excluding the base ? (Assume that all triangular faces are equal in area)
(A) 4250
(B) 6500
(C) 1500
(D) 3600
(E) 1625

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here is my solution with diagram. sorry for late
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B

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mikemcgarry
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In a particular 4-sided pyramid, each side of the square bas 15 Aug 2013, 15:34
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Asifpirlo
In a particular 4-sided pyramid, each side of the square base is 50 feet in length. If the apex of the pyramid is 60 feet from the ground, what is the total surface area of the pyramid, excluding the base ? (Assume that all triangular faces are equal in area)
(A) 4250
(B) 6500
(C) 1500
(D) 3600
(E) 1625
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A very nice problem! Congrats to whomever created this. :-)

Here's a solution. We know the base square is 50 x 50. We know the height, from center of the square base to the apex, is 60 feet. Imagine a right triangle that has the following three sides
(a) from the center of the square base perpendicularly to midpoint of an edge = 25 feet
(b) the height, from center of the square to the apex = 60 feet
(c) height down the center of a face triangle, from apex to the midpoint of an edge of the square base
Clearly, for this a & b & c, (a^2) + (b^2) = (c^2) applies, the Pythagorean Theorem.

Now, on the GMAT, with no calculator at your disposal, the most boneheaded thing in the world you could do at this point would be to square 25, then square 60, then add them, and then try, somehow, to extract the square root of the resulting four-digit number. Good luck with that! Instead, use proportional thinking. Both 25 and 60 are divisible by 5, so scale everything down by a factor of five. In this scaled-down world, a' = 5, b' = 12, and we can square those and find c --- in fact, if you know your Pythagorean Triplets, you may immediately recognize the {5, 12, 13} combination. If that's not familiar, see this post:
https://magoosh.com/gmat/2012/pythagorea ... -the-gmat/
So, in the scaled down world, c' = 13, so we scale back up to the pyramid world, and c = 5*13 = 65. That's the height of a face triangle. The base of the face triangle is 50, the edge of the square base. For one face triangle:

A = (0.5)b*h = (0.5)(50)*(65) = 25*65 = 1625

We have four of those triangles, so

Total area = 4*1625 = 6500

Answer choice (B).

Does all this make sense?

Mike :-)
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Asifpirlo
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In a particular 4-sided pyramid, each side of the square bas 16 Aug 2013, 04:14
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mikemcgarry
Asifpirlo
In a particular 4-sided pyramid, each side of the square base is 50 feet in length. If the apex of the pyramid is 60 feet from the ground, what is the total surface area of the pyramid, excluding the base ? (Assume that all triangular faces are equal in area)
(A) 4250
(B) 6500
(C) 1500
(D) 3600
(E) 1625
A very nice problem! Congrats to whomever created this. :-)

Here's a solution. We know the base square is 50 x 50. We know the height, from center of the square base to the apex, is 60 feet. Imagine a right triangle that has the following three sides
(a) from the center of the square base perpendicularly to midpoint of an edge = 25 feet
(b) the height, from center of the square to the apex = 60 feet
(c) height down the center of a face triangle, from apex to the midpoint of an edge of the square base
Clearly, for this a & b & c, (a^2) + (b^2) = (c^2) applies, the Pythagorean Theorem.

Now, on the GMAT, with no calculator at your disposal, the most boneheaded thing in the world you could do at this point would be to square 25, then square 60, then add them, and then try, somehow, to extract the square root of the resulting four-digit number. Good luck with that! Instead, use proportional thinking. Both 25 and 60 are divisible by 5, so scale everything down by a factor of five. In this scaled-down world, a' = 5, b' = 12, and we can square those and find c --- in fact, if you know your Pythagorean Triplets, you may immediately recognize the {5, 12, 13} combination. If that's not familiar, see this post:
https://magoosh.com/gmat/2012/pythagorea ... -the-gmat/
So, in the scaled down world, c' = 13, so we scale back up to the pyramid world, and c = 5*13 = 65. That's the height of a face triangle. The base of the face triangle is 50, the edge of the square base. For one face triangle:

A = (0.5)b*h = (0.5)(50)*(65) = 25*65 = 1625

We have four of those triangles, so

Total area = 4*1625 = 6500

Answer choice (B).

Does all this make sense?

Mike :-)
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Absolutely brilliant........Amazing.
just 1 reply... :-D
Thanks
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In a particular 4-sided pyramid, each side of the square bas 15 Sep 2020, 04:51
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Asifpirlo
In a particular 4-sided pyramid, each side of the square base is 50 feet in length. If the apex of the pyramid is 60 feet from the ground, what is the total surface area of the pyramid, excluding the base ? (Assume that all triangular faces are equal in area)
(A) 4250
(B) 6500
(C) 1500
(D) 3600
(E) 1625

Show Spoiler
here is my solution with diagram. sorry for late
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the total surface area of the pyramid, excluding the base=Lateral surface area

lateral surface area=\(\frac{1}{2}*P*L\) where, P = perimeter of the base, L= slant height(i.e PX in the attachment provided)

\(P=50*4=200\)

\(L=\sqrt{25^2 + 60^2}\)= 65

\(Area= \frac{1}{2} * 200*65=6500\) (Answer)
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