Friday
Friday: Slices and Spins
Comet turns the foam cone in her hands. "If we cut this straight across, the face is a circle. Smaller than the base."
"And down through the tip?" Wren asks. "What do you notice?"
"A triangle. Same cone, two different cuts, two different shapes."
Nova projects a rectangle spinning fast about one edge. The blur is a cylinder. "Would you like a hint? Spin, do not cut."
"A right triangle spun on a leg is a cone," Comet says. "A half circle spun on its diameter is a ball."
"Slices go from solid to flat. Spins go from flat to solid," Wren says. "Review time. Then next week we model the whole set."
"Our designer," Comet says, "every slice and every spin, then the week in five lines."
Cross-sections: solid to flat
A plane slicing a solid leaves a flat face, the cross-section. The shape depends on the solid and the direction of the cut.
| Solid | Cut | Cross-section |
|---|---|---|
| cylinder | parallel to the base | a circle |
| cylinder | through the axis | a rectangle |
| cone | parallel to the base | a smaller circle |
| cone | through the apex | a triangle |
| sphere | any plane | a circle |
| square pyramid | parallel to the base | a smaller square |
| cube | parallel to a face | a square |
Solids of rotation: flat to solid
- A rectangle spun about one of its sides sweeps out a cylinder.
- A right triangle spun about one of its legs sweeps out a cone.
- A half circle spun about its diameter sweeps out a sphere.
- A circle spun about a line outside it sweeps out a ring.
Volumes in a day
- A can of paint, a mug, a drum: cylinders. Base area times height.
- A paper party hat or a funnel: a cone, one third of its cylinder.
- A ball of any size: a sphere, 4/3 πr³.
- A tent with a square floor rising to a point: a pyramid, one third of its prism.
The week in five lines
- Wedges in a row show a circle's area is πr². Stacked discs show a cylinder is πr²h.
- Three pyramids fill a cube, and three cone-fulls fill a cylinder: Bh/3 and πr²h/3.
- A sphere is 4/3 πr³. Use 3.14 for π.
- Scaling every length by k multiplies the volume by k³.
- A slice leaves a cross-section. A spin sweeps a flat shape into a solid.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
A paint can is a cylinder with radius 5 cm and height 12 cm. What is its volume in cubic centimeters? Use 3.14 for π.
A paper funnel is a cone with radius 6 cm and height 10 cm. What is its volume in cubic centimeters? Use 3.14 for π.| Week review | True or false? |
|---|---|
| A cone is one third of the cylinder with the same base and height. | ? |
| Slicing a sphere always gives a circle. | ? |
| Slicing a cylinder through its axis gives a circle. | ? |
| Spinning a right triangle about a leg sweeps out a cylinder. | ? |
| Tripling every length multiplies the volume by 27. | ? |
Week eleven done, designer. Tomorrow is Shop Day: three containers from home, modeled and measured.