Why this matters for FAST: Volume problems appear frequently on FAST. Students must know when to use each formula, correctly identify radius vs. diameter, and apply formulas to real-world contexts.
Students use the given diameter directly in the formula instead of dividing by 2 to get the radius first.
"ALWAYS check: Is this the radius or diameter? Radius goes from center to edge. Diameter goes all the way across. If given diameter, DIVIDE BY 2 to get radius before using any formula!"
Students use V = pi r squared h for cones (forgetting the 1/3) or V = pi r cubed for spheres (forgetting the 4/3).
"A cone is exactly 1/3 of a cylinder with the same base and height - that's why we multiply by 1/3. For a sphere, the 4/3 comes from calculus, but you can remember: sphere formula has 4/3 and r CUBED (three dimensions!)."
Students use r squared in the sphere formula or r cubed in cylinder/cone formulas.
"Cylinders and cones have a circular BASE (r squared) times HEIGHT (h). Spheres have NO height - they're the same in all directions, so we use r CUBED. Remember: Base times height = r squared h. All directions = r cubed."
Review area of a circle: A = pi r squared. "This will be the BASE of our cylinders and cones!" Also review that volume is 3D - measured in cubic units.
"A cylinder is like stacking circles on top of each other. Volume = Base area times Height. Since the base is a circle, V = pi r squared h."
Cylinder: V = pi r squared h
r = radius of the circular base
h = height of the cylinder
Example: r = 3 cm, h = 10 cm
V = pi (3 squared)(10) = pi (9)(10) = 90 pi = 282.74 cubic cm
Cone: V = (1/3) pi r squared h
A cone is exactly 1/3 of a cylinder with the same base and height!
Example: r = 3 cm, h = 10 cm
V = (1/3) pi (3 squared)(10) = (1/3)(90 pi) = 30 pi = 94.25 cubic cm
Sphere: V = (4/3) pi r cubed
No height needed - a sphere is the same in all directions
Example: r = 3 cm
V = (4/3) pi (3 cubed) = (4/3) pi (27) = 36 pi = 113.10 cubic cm
Work through examples together:
"A cone and cylinder have the same radius (5 cm) and height (12 cm). How do their volumes compare?"
Correct answer: The cone's volume is 1/3 of the cylinder's volume. Cylinder V = pi (25)(12) = 300 pi. Cone V = 100 pi. You can fill the cone 3 times to equal the cylinder!
For struggling students: Focus on one shape at a time. Provide formula cards. Emphasize checking radius vs diameter. Use round numbers first.
For advanced students: Challenge with composite figures (hemisphere on cylinder, cone on cylinder). Have them derive why cone is 1/3 of cylinder experimentally.
For home: Send Parent Activity sheet. Families can measure cylindrical containers and calculate volume.