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).
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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
