Find the ratio of the volumes of (a) two similar solid cylinders of circumferences 10cm and 8 cm; (b) two similar solid cones of heights 9cm and 12cm; (c) two spheres of radii 4 cm and 6 cm.
If you have two similar figures then the ratio of corresponding lengths is a constant called the scale factor. If the scale factor between two similar figures is a:b, then the ratio of any corresponding lengths is a:b. The ratio of any corresponding areas is `a^2:b^2` and the ratio of any corresponding volumes is `a^3:b^3` .
(a) If the ratio of circumferences (lengths) is 10:8=5:4 then the ratio of volumes is `5^3:4^3` or `125:64`
(b) If the ratio of heights (lengths) is 9:12=3:4 then the ratio of volumes is `3^3:4^3=27:64`
(c) If the ratio of radii (lengths) is 4:6=2:3 then the ratio of volumes is `2^3:3^3=8:27`
This last is easy to verify. The volume of a sphere is `V=4/3pir^3`
So the volume of a sphere with radius 4 is `4/3pi4^3=(256pi)/3`
The volume of a sphere with radius 6 is `4/3pi6^3=288pi`
The ratio of the volumes is `((256pi)/3)/(288pi)=256/864=8/27`