Cylinder surface area
The wall is an unrolled rectangle whose width is the circle’s circumference and whose height is the cylinder height. Count included circular ends separately
\[SA=2\pi rh+n\pi r^2\]
- Choose n Use two for both ends, one for one end, or zero for neither end
- Convert and recover radius Use one length unit and halve a given diameter
- Final precision Keep π in the combined expression, then write the requested number of decimal places
Worked example
An open-top cylinder has radius 2 cm and height 7 cm. Find its surface area to 1 decimal place
\[r=2\text{ cm},\quad h=7\text{ cm}\]
- \(\displaystyle A_{\text{wall}}=2\pi(2)(7)=28\pi\)Unroll the wall into a rectangle
- \(\displaystyle A_{\text{base}}=\pi(2)^2=4\pi\)There is one included circular end
- \(\displaystyle SA=32\pi\approx100.5\text{ cm}^2\)Combine first and round once
Common mistakes
Adding the absent top
✕\(\displaystyle SA_{\text{open}}=2\pi r^2+2\pi rh\)✓\(\displaystyle SA_{\text{open}}=\pi r^2+2\pi rh\)
The open top is not an included surfaceUsing diameter as radius
✕\(\displaystyle r=d\)✓\(\displaystyle r=d/2\)
The diameter crosses the full circleRounding every part first
✕\(\displaystyle \begin{gathered}SA\approx\operatorname{round}(A_1)\\{}+\operatorname{round}(A_2)\end{gathered}\)✓\(\displaystyle SA\approx\operatorname{round}(A_1+A_2)\)
Use the exact sum before the final roundingDifficulty levels
standardFind the total area of a closed cylinder
challengeFind the area of the wall and base of an open-top cylinder
Questions about this topic
Why does the wall use circumference?+
Unwrapping the side preserves the distance around the circle, which becomes the rectangle’s width
Can I halve the closed area for an open cylinder?+
No. Subtract only the absent circular end
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
5A Length conversion and perimeter: Convert measurement units5A Length conversion and perimeter: Perimeter and area of standard shapesFind an unknown side from a perimeter5A Length conversion and perimeter: Solve a linear equation5A Length conversion and perimeter: Substitute values into a formula5A Length conversion and perimeter: Round measurements to decimal places5B Circumference, sectors and exact π: Circumference and area of a circle5B Circumference, sectors and exact π: Arc length and sector areaPerimeter of a sector5B Circumference, sectors and exact π: Simplify exact expressions involving pi5B Circumference, sectors and exact π: Substitute values into a formula5B Circumference, sectors and exact π: Round measurements to decimal places5C Area formulas and units: Convert measurement units5C Area formulas and units: Perimeter and area of standard shapes5C Area formulas and units: Circumference and area of a circle5C Area formulas and units: Arc length and sector area5C Area formulas and units: Simplify exact expressions involving pi5C Area formulas and units: Substitute values into a formula5C Area formulas and units: Round measurements to decimal placesPerimeter and area of composite shapes5D Composite perimeter and area: Find a side of a right-angled triangle5D Composite perimeter and area: Substitute values into a formula5D Composite perimeter and area: Round measurements to decimal places5E Right-prism surface area: Surface area of cuboidsSurface area of a triangular prism5E Right-prism surface area: Find a side of a right-angled triangle5E Right-prism surface area: Substitute values into a formula5E Right-prism surface area: Round measurements to decimal placesPyramid surface area: Surface area of square pyramids (optional extension)5F Cylinder surface area: Surface area of cylindersSurface area of a cylinder portion5F Cylinder surface area: Substitute values into a formula5F Cylinder surface area: Round measurements to decimal places5G Volume and capacity: Convert measurement units5G Volume and capacity: Volume of right prisms5G Volume and capacity: Capacity of a container5G Volume and capacity: Solve a linear equation5G Volume and capacity: Substitute values into a formula5G Volume and capacity: Round measurements to decimal places5H Cylinder volume and capacity: Volume of cylinders5H Cylinder volume and capacity: Capacity of cylindersVolume of a cylinder portion5H Cylinder volume and capacity: Substitute values into a formula5H Cylinder volume and capacity: Round measurements to decimal placesSmallest unit and limits of accuracy