Similar Pyramids and Measurement RatiosDate: 08/10/2002 at 21:12:19 From: Deanna Subject: Similar Pyramids Dear Dr. Math, The volumes of two similar pyramids are 27 and 64. If the smaller has lateral surface area of 18, how would I find the lateral surface area of the larger one? I have tried to go backward since V=1/3bh, so, 27/3 = 9, but I don't know what I'm really doing. I'm confusing myself the more I think about it. Can you please walk me through this? Thank you for taking time to read this. Sincerely, Deanna Date: 08/11/2002 at 02:42:11 From: Doctor Greenie Subject: Re: Similar Pyramids Hi, Deanna - You COULD probably get to the answer the way you are going, but it would be a LOT of work, and it is work you don't need to do. You can solve this problem using a very useful fact about similar figures that is stressed far too seldom in the math curriculum. The fact is this: If the scale factor (ratio between linear measurements of two similar figures) is a:b, then (1) the ratio between any area measurements of those two similar figures will be a^2:b^2 and (2) the ratio between any volume measurements of those two similar figures will be a^3:b^3 In other words, if the ratio of any measurement in one dimension between two similar figures is a:b, then the ratio of any measurement in two dimensions between those two similar figures is a^2:b^2, and the ratio of any measurement in three dimensions between those two similar figures is a^3:b^3. This principle works 'in any direction'. If you know that the ratio of some area measurement between two similar figures is a^2:b^2, then you know that the ratio between any two linear measurements will be a:b, and that the ratio between any two volume measurements will be a^3:b^3. And - appropriate to your case - if you know that the ratio of some volume measurement between two similar figures is a^3:b^3, then you know that the ratio between any two linear measurements will be a:b, and that the ratio between any two area measurements will be a^2:b^2. You are told that the volumes of the two similar pyramids are 27 and 64. The ratio of these two volume measurements is 27:64, which is 3^3:4^3. You therefore know that the ratio between any linear measurements on the two similar pyramids will be 3:4, and that the ratio between any area measurements on the two similar pyramids will be 3^2:4^2 = 9:16. You are told that the lateral surface area of the smaller pyramid is 18; to find the lateral surface area of the larger pyramid, we just need to solve a proportion using the ratio we now know between area measurements on the two pyramids. So we have 9 18 -- = -- 16 x and we find that the surface area of the larger pyramid is 32. I hope this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 08/11/2002 at 22:46:23 From: Deanna Subject: Similar Pyramids Dr. Math, So would the conclusion be that since the volumes of the triangles are 27 and 64, the lateral surface areas are 18 and 32? Thank you so much for helping me. Deanna Date: 08/12/2002 at 01:21:49 From: Doctor Greenie Subject: Re: Similar Pyramids Hi, Deanna -- Yes, that's right. Thanks for taking the time to send us your thanks. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ |
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