Geometry / Unit 11: Area, Surface Area and Volume
11.7 Similar Solids and Cross Sections
Notes
- Similar solids with scale factor $k$ have surface area ratio $k^2$ and volume ratio $k^3$.
- A cross section is the shape you get by slicing a solid with a plane.
- Slicing a cylinder parallel to its base gives a circle; perpendicular to its base (through the axis) gives a rectangle.
- Rotating a 2D shape around a line makes a solid: a rectangle spun around one side makes a cylinder; a right triangle spun around a leg makes a cone.
Practice
Two similar boxes have a scale factor of $1:2$. The smaller holds 15 cubic inches. How much does the larger hold?
Volume scales by $2^3 = 8$.
A cone is sliced by a plane parallel to its base. What is the cross section?
Slices parallel to the base of a cone are circles.
A right triangle with legs 3 and 4 is rotated around the leg of length 4. Which solid is formed, and what is its volume?
It forms a cone with $r = 3$ and $h = 4$: $\displaystyle \frac{1}{3}\pi(9)(4) = 12\pi$.
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