Surface area of a cylinder portion practice questions | Year 9 Mathematics

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

\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

✕\(\displaystyle \begin{gathered}SA=\frac12(18\pi+48\pi)\\{}=33\pi\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=\frac12(18\pi+48\pi)+48\\{}=(33\pi+48)\text{ cm}^2\end{gathered}\)

The original surfaces are halved, but the cut creates a new flat face that must also be counted

Leaving out the cut face

✕\(\displaystyle SA=56\pi\text{ cm}^2\)✓\(\displaystyle SA=(56\pi+80)\text{ cm}^2\)

The flat face made by the cut is an included surface of the closed half-cylinder

Ignoring the portion fraction

✕\(\displaystyle \begin{gathered}SA=32\pi+80\pi+80\\{}=(112\pi+80)\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}SA=\frac14(32\pi+80\pi)+80\\{}=(28\pi+80)\text{ cm}^2\end{gathered}\)

Only the stated fraction of the curved wall and end areas belongs to the portion

Difficulty 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 9 Mathematics, chapter 5: Measurement

In this chapter