Mathematics 7–10 · Year 10
Cone and cylinder of equal base and height: three pours
Measurement and space
The idea
A cone fills a cylinder of the same base and height exactly three times, which is the 1/3 in V = (1/3)πr²h.
What you need
- Set of hollow relational solids: cone and cylinder with the same base radius and height (a commercial relational-solids set, or made from card and sealed with tape; the worked figures use r = 4.0 cm and h = 9.0 cm)
- Dry rice (water only with a waterproof plastic set), tray, funnel
- Ruler and calculator
How to do it
- Measure r and h of both solids and confirm they match; compute πr²h for the cylinder.
- Fill the cone level with rice and pour it into the cylinder; repeat, counting pours until the cylinder is level full.
- Compute the cone's volume as the cylinder's volume divided by the number of pours and compare with (1/3)πr²h.
- If a hemisphere of the same radius is in the set, mark a line inside the cylinder at height 2r (8.0 cm for r = 4.0 cm), count how many hemisphere pours fill it to that line (three), and so find the sphere to be two thirds of a cylinder of height 2r.
What you should see
For r = 4.0 cm and h = 9.0 cm the cylinder holds 452.4 cm³ and the cone 150.8 cm³; the cone fills the cylinder in exactly three level pours, with the third pour finishing level to within a few millimetres of rice. A hemisphere of radius 4.0 cm (134.0 cm³) fills the same cylinder to the 8.0 cm line (402.1 cm³) in three pours, leaving the top 1.0 cm empty. The learner knows it worked when the third pour finishes with neither a gap nor an overflow beyond about 3 per cent.
What changes
This activity lists no variables to change, measure and keep the same.
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- A cone holds half the cylinder.
- The 1/3 comes from the slanted side, so it changes with the slope.
- Volume formulas for curved solids cannot be checked by hand.
Safety card
Hazards
- spilt rice on the floor is a slip hazard
Controls
- pour over a tray
- sweep spills at once
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Volume B (Path); page read 2026-09-22. acara_v9 is empty by decision, not by omission: ACARA v9 Year 10 Measurement has no description for the volume of a cone, pyramid or sphere, the whole set being AC9M10M01 to AC9M10M05, of which AC9M10M01 reaches only composite objects; v9 records re-read 2026-09-23MA5-VOL-P-01
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.