Eighth Grade · Math · week 33

Volume: cylinders, cones, and spheres

Apply the volume formulas for cylinders, cones, and spheres to solve real problems.

Three curved solids share the same building block, the circle-area formula pi times r^2. A cylinder's volume is that circle area stacked up: V = pi x r^2 x h. A cone is exactly one-third of the matching cylinder (same radius and height): V = (1/3) x pi x r^2 x h. A sphere has its own formula: V = (4/3) x pi x r^3.

Take pi to be about 3.14 for a decimal estimate, but keep the exact form (like 90 x pi) visible too -- both are correct, and knowing which is wanted matters.

📋 For the teacher
Teach it (10–15 min):
  • Anchor all three formulas to the same building block, pi x r^2 -- a cylinder stacks that circle up height times; a cone is one-third of the matching cylinder.
  • Take pi to be about 3.14 consistently this week for a decimal answer, but keep the exact form (like 90 x pi) visible too.
  • Watch order of operations closely: square (or cube) the radius FIRST, then multiply by everything else -- multiplying first is a common, formula-breaking mistake.
  • Real containers (cans, ice cream cones, balls) make every formula concrete -- measure a real object's radius and height if one is handy.
Talk about it:
  • Why does a cone's volume formula have that extra one-third compared to a cylinder with the same base and height?
  • What is the shared building block hiding inside all three formulas?
  • Why might it matter whether you give a volume answer as "90 x pi" or as "282.6"?

Mastery looks like: Given the radius (and height, where needed), they correctly compute the volume of a cylinder, cone, or sphere, keeping order of operations correct.

Common stumbles: Multiplying the radius by other numbers before squaring or cubing it; forgetting the one-third factor for cones or the four-thirds factor for spheres.

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.
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