Volume of a cylinder portion

Radius

\(\displaystyle 8\text{ cm}\)

Length

\(\displaystyle 25\text{ cm}\)

Sector angle

\(\displaystyle 90^\circ\)

A quarter-cylinder has radius 8 cm and length 25 cm. What is its volume to 1 decimal place?

Radius and cylinder length stay fixed. The sector angle changes from 90° to 180°. What happens to the portion volume?

Change sector angle and radius while cylinder length stays 10 cm. Both bars show volume in cm³. Values are rounded for display; the fraction uses the exact angle

Which relationship holds while the other dimensions remain fixed?

Sector area extended through a length

A portion is a sector-shaped cross-section extended straight along the cylinder. Its volume uses the sector fraction; surface cut faces are not added

1. 1Use exact fractions Half and quarter cylinders use 1/2 and 1/4; general sectors use θ/360.
2. 2Exact coefficient When V = bπ cubic units is requested, enter b only, as an exact integer, decimal or fraction
3. 3Capacity Use internal dimensions, calculate the exact portion volume, then convert to litres or millilitres before final rounding

Radius

\(\displaystyle 1.5\text{ cm}\)

Length

\(\displaystyle 10\text{ cm}\)

Exact form

\(\displaystyle V=b\pi\text{ cm}^3\)

A half-cylinder has radius 1.5 cm and length 10 cm. Its exact volume is bπ cm³. Enter b only

Angle

\(\displaystyle 270^\circ\)

Diameter

\(\displaystyle 12\text{ cm}\)

Length

\(\displaystyle 20\text{ cm}\)

A cylinder portion has sector angle 270°, diameter 12 cm and length 20 cm. Find its volume to 2 decimal places

Internal diameter

\(\displaystyle 50\text{ cm}\)

Internal length

\(\displaystyle 120\text{ cm}\)

A container’s usable interior is a half-cylinder of internal diameter 50 cm and length 120 cm. Find its capacity to 1 decimal place

Containers, culture and mathematical models

Museums Victoria documents a no-bintarno, a wooden drinking vessel from the Melbourne area, used to scoop water. Its curved, irregular interior is not a perfect cylinder. This example connects the design of First Peoples’ containers with the distinction between surface area, volume and usable capacity

1. 1Investigate the object The catalogue describes the vessel and its materials. Object X 1561, Museums Victoria: https://collections.museumsvictoria.com.au/items/158691
2. 2State modelling limits The next two half-cylinder models are invented mathematical examples, not measurements or replicas of the museum object. Curved ends, wall thickness and irregular shape limit a simple model’s accuracy
3. 3Compare quantities An open half-cylinder model has inside area πrL + πr² and usable volume ½πr²L. The open flat top is not an inside surface

Model A

\(\displaystyle r=4\text{ cm},\quad L=20\text{ cm}\)

Model B

\(\displaystyle r=6\text{ cm},\quad L=10\text{ cm}\)

Inside area, no flat top

\(\displaystyle A=\pi rL+\pi r^2\)

Internal volume

\(\displaystyle V=\frac12\pi r^2L\)

Compare two hypothetical open half-cylinder containers with semicircular ends. Model A has internal radius 4 cm and length 20 cm; model B has internal radius 6 cm and length 10 cm. Which statement correctly connects their inside surface areas and capacities?

A cylinder portion has angle 120°, radius 3 cm and length 10 cm. Which line contains the first error?