IM Geo.5.6 Lesson: Scaling Solids
Find the volume of each cube mentally.
Use the applet to build cubes that result from dilating a unit cube by each scale factor shown in the table. Then, complete the table with the surface area and volume of each dilated cube.
Suppose a unit cube is dilated by some scale factor k.
Write an expression for the surface area of the dilated cube.
Write an expression for the volume of the dilated cube.
Compare and contrast the expression for surface area and the expression for volume.
Clare says, “We know that if we dilate a cube by a factor of , the cube’s volume is multiplied by . It seems like that must apply to all solids, but I’m not sure how to prove it.” Elena says, “Earlier in the unit, we showed that we can cover any two-dimensional shape with rectangles, so the property that area changes by when we dilate a figure by applies to all shapes, not just rectangles. Can we do something similar here?” Use Elena’s line of reasoning to argue that for any solid, if it’s dilated by a factor of , the volume is multiplied by .
Suppose a triangular prism has surface area 84 square centimeters and volume 36 cubic centimeters. The prism is dilated by scale factor . Calculate the surface area and volume of the dilated prism.
The image shows a figure called Sierpinski’s triangle.
It’s formed by starting with an equilateral triangle, then repeatedly removing equilateral triangles created by joining the midpoints of the existing triangle’s sides. The first few stages are shown.
If we continue this process of removal forever, we are left with some points that never get removed from the triangle. The remaining points are what we call Sierpinski’s triangle. At any given stage, the triangle at the top of the figure is a scaled copy of the triangle at the previous stage.
stage 1 stage 2 stage 3 
For the completed Sierpinski’s triangle, though, the top triangle is a scaled copy not of the previous stage, but of the full Sierpinski’s triangle.
For the completed figure, what scale factor takes Sierpinski’s triangle to its scaled copy at the top?
stage 4 top triangle 
Based on the scale factor, what fraction of the original shaded region should be contained in the scaled copy at the top?
The scaled copy at the top actually contains of the shading of the original. Provide reasoning that shows that this is true.