Same Math › Geometry · grades 9–10 › Area and volume formulas
How do kids remember volume formulas?
Most volume formulas are one idea: area of the base × height. A prism or cylinder is base × height (a cylinder's base is a circle, so V = πr²h). A pyramid or cone that comes to a point is one third of that (V = ⅓πr²h for a cone). A sphere is the odd one out: V = (4/3)πr³. Kids find the base area first, then multiply.
- Cylinder: circle base × height.
- Cone: one third of the cylinder.
- r is the radius (half the diameter); h is straight up.
Why schools teach it
One idea instead of six formulas
Eighth graders use the formulas for cylinders, cones and spheres (8.G.C.9), and Geometry students explain where they come from (HSG.GMD.A.3), like why a cone is exactly one third of a cylinder.
Seeing that prisms and cylinders share base × height, and pyramids and cones share one third of it, cuts the memorizing down to the base areas and the sphere.
At the kitchen table
What to say, and what to skip
Say this
What shape is the base?
Find its area first.Is it the radius or the diameter?
Halve the diameter.Does the shape come to a point?
Then take one third.Leave it in terms of π, or round?
Check the directions.
Skip this
- Using the diameter as r.A 10-inch-wide cylinder has r = 5.
- Slant height for volume.Volume uses the straight-up height; slant height is for surface area.
- Square units for volume.Volume is cubic units: cm³, in³.
See it in action
A cone with radius 3 cm and height 8 cm
Base area
π · 3² = 9πTimes height, then one third
⅓ · 9π · 8 = 24πRound if asked
24π ≈ 75.4 cm³
A cylinder the same size would hold 72π, exactly three times as much.
Free practice sheet
Make a volume sheet
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Parents also ask
Why is a cone one third of a cylinder?
It can be shown with water or sand: three cones fill the matching cylinder. Calculus proves it.
Where does the sphere formula come from?
A sphere is two thirds of the cylinder that just fits around it. Kids usually just learn the formula.
What's the difference between volume and surface area?
Volume is how much fits inside (cubic units); surface area is the wrapping paper (square units).
Should they use 3.14 or the π button?
The π button unless the teacher says 3.14. The answers differ a little.