Surface area of a cylinder portion
A cylinder portion contains a fraction of the full curved wall and circular ends, together with any flat faces made by the cut. Add each included face group once and round only the final total. Here F is the total area of included flat cut faces. The rules below use a closed half-cylinder with radius 4 cm and length 10 cm
\[\begin{gathered}f=\frac{\theta}{360}\\SA=2\pi rf(h+r)+F\end{gathered}\]
- Find the portion fraction Divide the cross-section angle by 360 degrees \(\displaystyle f=\frac{180}{360}=\frac{1}{2}\)
- Find the curved wall Multiply the full cylinder wall area by the portion fraction \(\displaystyle \begin{gathered}C=\frac12\times2\pi(4)(10)\\{}=40\pi\text{ cm}^2\end{gathered}\)
- Find the end faces Multiply the full pair of circular ends by the same fraction \(\displaystyle \begin{gathered}E=\frac12\times2\pi(4)^2\\{}=16\pi\text{ cm}^2\end{gathered}\)
- Count the cut face For a closed half-cylinder, add the rectangle with width 2 r. A sector portion other than a half has two rectangles r by h. Round the total once \(\displaystyle \begin{gathered}F=2(4)(10)=80\text{ cm}^2\\SA=(56\pi+80)\text{ cm}^2\end{gathered}\)
Worked example
A closed quarter-cylinder has radius 4 m and length 10 m. Find its total surface area to 2 decimal places
\[\begin{gathered}r=4\text{ m},\quad h=10\text{ m}\\\theta=90^\circ\end{gathered}\]
- \(\displaystyle f=\frac{90}{360}=\frac{1}{4}\)A 90-degree cross-section is one quarter of a full turn
- \(\displaystyle \begin{gathered}C=\frac14\times2\pi(4)(10)\\{}=20\pi\text{ m}^2\end{gathered}\)Find the quarter of the full curved wall
- \(\displaystyle \begin{gathered}E=\frac14\times2\pi(4)^2\\{}=8\pi\text{ m}^2\\F=2(4)(10)=80\text{ m}^2\end{gathered}\)Find the quarter of the two ends and the two radial cut faces
- \(\displaystyle \begin{gathered}SA=20\pi+8\pi+80\\{}=(28\pi+80)\text{ m}^2\\{}\approx167.96\text{ m}^2\end{gathered}\)Add all included surfaces and round the final total to 2 decimal places
Common mistakes
Halving the full surface area: half-cylinder, radius 3 cm, length 8 cm
✕\(\displaystyle \begin{gathered}SA=\frac12(18\pi+48\pi)\\{}=33\pi\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=\frac12(18\pi+48\pi)+48\\{}=(33\pi+48)\text{ cm}^2\end{gathered}\)
The original surfaces are halved, but the cut creates a new flat face that must also be countedLeaving out the cut face: half-cylinder, radius 4 cm, length 10 cm
✕\(\displaystyle SA=56\pi\text{ cm}^2\)✓\(\displaystyle SA=(56\pi+80)\text{ cm}^2\)
The flat face made by the cut is an included surface of the closed half-cylinderIgnoring the portion fraction: quarter-cylinder, radius 4 cm, length 10 cm
✕\(\displaystyle \begin{gathered}SA=32\pi+80\pi+80\\{}=(112\pi+80)\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=\frac14(32\pi+80\pi)+80\\{}=(28\pi+80)\text{ cm}^2\end{gathered}\)
Only the stated fraction of the curved wall and end areas belongs to the portionDifficulty levels
introAccount for a closed half-cylinder and give an exact total
standardAccount for a closed quarter-cylinder and round the total to a stated number of decimal places
challengeHandle a general sector angle, a stated diameter, an open face, or an exact total
examChoose the rounded total from options containing fraction, cut-face, and half-surface errors
Questions about this topic
Why is a cut face added?+
Slicing a cylinder creates a new flat surface, and that surface is included when the object is closed or exposed
How are the curved wall and ends scaled?+
Each is multiplied by the cross-section angle divided by 360 because the portion contains that fraction of the full cylinder surface
Why can the surface area not simply be halved?+
Halving the original surfaces does not include the new rectangular cut face
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement
In this chapter
Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)