Unit 12: Solids: Surface Area, Volume, and Cross-Sections
Wrapping and filling three-dimensional shapes, the slices hiding inside them, and the spinning trick that turns a flat shape solid.
10 topics · 34 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Nets and Surface Area of Prisms and Cylinders
- How it worksHow does a cylinder's curved surface unroll into a flat rectangle to find its area?
- Worked exampleFind the surface area of a rectangular prism with length 5, width 4, and height 3.
- Worked exampleFind the surface area of a cylinder with radius 4 and height 10.
- Common mistakeWhy doesn't a cylinder's surface area formula include a term for its diameter directly?
Your turn1 of 2
Find the surface area of a rectangular prism with length 15, width 14, and height 19.Surface Area of Pyramids and Cones
- How it worksHow does a pyramid's slant height, not its vertical height, determine the area of each triangular face?
- Worked exampleFind the surface area of a square pyramid with base side 6 and slant height 5.
- Worked exampleFind the surface area of a cone with radius 3 and slant height 7.
Your turn1 of 2
Find the surface area of a square pyramid with base side 15 and slant height 14.Volume of Prisms and Cylinders
- How it worksHow does multiplying a base's area by the solid's height give the volume of any prism or cylinder?
- Worked exampleFind the volume of a triangular prism with a triangular base of area 12 and a height of 9.
- Worked exampleFind the volume of a cylinder with radius 5 and height 12.
- Common mistakeWhy doesn't tripling a cylinder's radius only triple its volume the way tripling its height would?
Your turn1 of 2
Find the volume of a prism with a base area of 35 and a height of 20.Volume of Pyramids and Cones
- How it worksHow do three congruent pyramids fit together to exactly fill one cube?
- Worked exampleFind the volume of a square pyramid with base side 6 and height 10.
- Worked exampleFind the volume of a cone with radius 4 and height 9.
- Common mistakeWhy isn't a cone's volume just its base area times its height, the way a cylinder's is?
Your turn1 of 2
Find the volume of a square pyramid with base side 15 and height 20.Volume and Surface Area of a Sphere
- How it worksHow does peeling a sphere into thin circular slices lead to its volume formula?
- Worked exampleFind the volume of a sphere with radius 6.
- Worked exampleFind the surface area of a sphere with radius 9.
- Going furtherWhat happens to a sphere's volume compared to its surface area as its radius keeps growing?
Your turn1 of 2
Find the volume of a sphere with radius 15.Cavalieri's Principle
- How it worksHow does matching cross-sectional area at every height guarantee two solids share the same volume?
- Worked exampleFind the volume of an oblique cylinder with base radius 4 and height 10, the same dimensions as a right cylinder.
- Ties togetherHow does Cavalieri's principle justify using the same volume formula for both a right prism and a slanted one?
Your turn1 of 2
Find the volume of an oblique cylinder with base radius 11 and height 20 (the perpendicular distance between its two bases).Cross-Sections of Solids
- How it worksHow does the angle and position of a slicing plane determine the exact shape of a solid's cross-section?
- Worked exampleFind the shape of the cross-section made by slicing a cone with a plane parallel to its base.
- Going furtherWhat happens to a cone's cross-section as the slicing plane tilts from parallel to the base toward parallel to the side?
Your turn1 of 2
A cone is sliced by a plane parallel to its base, partway up. What shape is the resulting cross-section?Solids of Revolution
- How it worksHow does rotating a right triangle around one of its legs sweep out a cone?
- Worked exampleFind the volume of the cone generated by rotating a right triangle with legs 3 and 8 around its 8 unit leg.
- Common mistakeWhy doesn't rotating a shape around a line that touches it produce the same solid as rotating it around a line far away?
Your turn1 of 2
A right triangle with legs 11 and 14 is rotated around its 14-unit leg. Find the volume of the resulting cone.Similar Solids and Scale Factor in 3D
- How it worksHow do a scale factor's square and cube separately predict a similar solid's surface area and volume?
- Worked exampleFind the volume of a similar solid scaled by a factor of 3 if the original solid's volume is 20.
- Common mistakeWhy doesn't doubling a solid's dimensions only double the amount of material needed to build it?
Your turn1 of 2
Find the volume of a similar solid scaled by a factor of 4, if the original solid's volume is 35.Optimizing Shape to Minimize Material
- How it worksHow does a cylinder's surface area change shape as its radius and height trade off for the same fixed volume?
- Worked exampleFind the surface area of two different cylinders that both hold a volume of 355 cubic centimeters, one short and wide and one tall and narrow.
- Going furtherWhat happens to the minimum-material shape question when the container has to be a rectangular box instead of a cylinder?
Your turn1 of 2
A cylinder must hold a fixed volume of 712 cubic units with a radius of 7. Find the height needed.