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# 3d geo

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Let APQRS be a pyramid, where the base PQRS is a square of side length 30. The total surface area of pyramid APQRS (including the base) is 1100. Let W, X, Y, and Z be the midpoints of AP, AQ, AR, and AS respectively. Find the total surface area of frustum PQRSWXYZ (including the bases).

Jan 12, 2021

#1
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The base of the original pyramid=   30^2 = 900

The sides of  the pyramid  are  4 congruent triangles  with  bases of 30  and a slant height = S

We can figure the slant height of the original pyramid, S , as

Then the area of the sides  =  1100   - 900   = 200

Connecting the  midpoints  of    AP, AQ, AR. AS  will form a  similar pyramid to  the original

The scale factor of this pyramid to  the original  =  1/2

So.....the total   surface area of  this pyramid  =  (scale factor)^2  * 1100 =  1100/4 = 275

The base of  this new pyramid =  15^2  = 225

So the surface area of the 4 sides of this new pyramid = 275 - 225  = 50

So  the  sides of the  frustum created have a surface area of    200  - 50 =   150

So....the total surface  area of the  furstum  =

Area of bottom base  +  area of sides  + area of top base  =

Area of bottom base  + area of sides +  area of the base of the  new pyramid =

900  +  150  +   225   =

1275  units^2

Jan 12, 2021
#2
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Hello, Guest!

Such a pyramid is impossible!!!

The total slant area of a pyramid cannot be smaller than the area of its base!!!

Make sure that your numbers are correct.

Jan 12, 2021
edited by jugoslav  Jan 12, 2021