Tenth Grade · Math · week 35

Surface area and volume of prisms, cylinders, pyramids, and cones

Apply the volume and surface area formulas for prisms, cylinders, pyramids, and cones, distinguishing vertical height from slant height.

Prisms and cylinders share one volume pattern: Volume = (area of the base) * height -- stack copies of the base, and the height tells how many "copies" tall the solid is. For a cylinder, the base is a circle, so Volume = pi * r^2 * h. Their surface areas share a pattern too: two congruent bases plus a lateral (side) area that unrolls into a rectangle. A prism's surface area is 2*(base area) + (perimeter of the base)*height; a cylinder's is 2*pi*r^2 + 2*pi*r*h, since its unrolled lateral surface is a rectangle with width equal to the circle's circumference.

Pyramids and cones share a different pattern -- exactly ONE THIRD of the prism or cylinder pattern for volume: Volume = (1/3) * (area of the base) * height (demonstrable by pouring: it takes exactly three pyramid-fuls, or three cone-fuls, to fill the matching prism or cylinder with the same base and height). Their surface areas both use a slant height l (the distance from the apex down the SLANTED outer surface to the base edge, not the straight-up vertical height): a regular pyramid's surface area is (base area) + (1/2)*(perimeter of the base)*l, and a cone's is pi*r^2 + pi*r*l.

📋 For the teacher
Teach it (10–15 min):
  • Use (or improvise with paper cones/cylinders and rice or sand) a physical pyramid/cone-into-prism/cylinder volume demonstration to literally pour and count "exactly 3."
  • Have them unroll a paper cylinder's label to physically see the lateral surface become a flat rectangle with one side equal to the circumference.
  • Draw a clear distinction on every pyramid or cone diagram between the vertical height and the slant height -- mixing these two is the single biggest error this week.
  • Group the eight formulas into the two families (prism/cylinder and pyramid/cone) on one reference sheet, so the "same pattern, different base shape" structure is visible at a glance.
Talk about it:
  • Why does a cone's volume come out to exactly one third of the matching cylinder's volume?
  • Why does a cone or pyramid's surface area use the SLANT height instead of the vertical height?
  • How is the "base times height" volume pattern for a prism similar to the area formula for a parallelogram?

Mastery looks like: They correctly apply the volume and surface area formulas for prisms, cylinders, pyramids, and cones, correctly distinguishing vertical height from slant height where needed.

Common stumbles: Using the vertical height instead of the slant height in a pyramid or cone's surface area formula; forgetting the 1/3 factor for pyramid and cone volumes.

Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
← Back to todayAll coursesYear planner