Volume of right prisms
Use the perpendicular cross-sectional area and prism length, not the total face area
\[V=AL,\quad L=V\div A\]
- Choose the area formula Use the shape of the cross-section, not every number in the diagram \(\displaystyle \begin{gathered}A_{\triangle}=\frac12 bh\\A_{\text{trapezium}}=\frac12(a+b)h\end{gathered}\)
- Composite solids Add the volumes of non-overlapping components. Subtract an empty or removed prism instead
- Dimensions and units Convert lengths before multiplying, or convert the final cubic measure using a cubed length factor
Common mistakes
Multiplying every given length
✕\(\displaystyle \begin{gathered}V=6\times8\times10\times15\\=7200\text{ cm}^3\end{gathered}\)✓\(\displaystyle \begin{gathered}V=\frac12\times6\times8\times15\\=360\text{ cm}^3\end{gathered}\)
The third side of the triangle is not used in its area formula, so it is not another volume factorDropping the triangular half
✕\(\displaystyle V=6\times8\times15=720\text{ cm}^3\)✓\(\displaystyle \begin{gathered}V=\frac12\times6\times8\times15\\=360\text{ cm}^3\end{gathered}\)
A triangular cross-section has area one-half times its base times its perpendicular height, so leaving out one-half doubles the volumeConfusing surface area and volume
✕\(\displaystyle \begin{gathered}SA=2\left(\frac12\times6\times8\right)\\{}+(6+8+10)\times15\\=408\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}V=\frac12\times6\times8\times15\\=360\text{ cm}^3\end{gathered}\)
The face sum has square units and finds surface area, while cross-sectional area times length has cubic units and finds volumeDifficulty levels
introRectangular-prism volume
standardTriangular and trapezoidal cross-section volume
challengeMissing length, given cross-sectional area and composite-prism volume
examRecognise the final triangular-prism volume
Questions about this topic
Can equal volumes have different surface areas?+
Yes. For example, closed prisms 2 × 2 × 16 cm and 4 × 4 × 4 cm both have volume 64 cm³, but surface areas 136 cm² and 96 cm²
Why do cubic units matter?+
Area times a perpendicular length has three length dimensions. A metre-to-centimetre factor of 100 becomes 100³ for volume
Which equal-volume prism has the largest surface area?+
Among the seven slider models, the 8 cm × 8 cm × 1 cm prism has the largest surface area, 160 cm². This is not a maximum for all prisms: a 1 cm × 1 cm × 64 cm prism has the same volume but surface area 258 cm². Without limits on dimensions, making a rectangular prism progressively thinner and longer gives no finite largest surface area
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