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Volume of prisms and cylinders

Start a set of 8
1 OF 18
Base of the end
\(\displaystyle 3\text{ m}\)
Height of the end
\(\displaystyle 2\text{ m}\)
Sloping sides of the end
\(\displaystyle 2.5\text{ m}\)
Length of the tent
\(\displaystyle 4\text{ m}\)

A ridge tent is a triangular prism. Its triangular end has base 3 m, height 2 m and sloping sides 2.5 m. The tent is 4 m long. How much space is inside the tent?

Year 10, Chapter 1, 4 difficulty levels, Calculator allowedBuild a worksheet

Volume of right prisms and cylinders

Use the perpendicular cross-sectional area and prism length, not the total face area

\[\begin{gathered}V=AL,\quad L=V\div A\\A_{\text{circle}}=\pi r^2\end{gathered}\]
  1. Choose the area formula Use the shape of the cross-section, not every number in the diagram \(\displaystyle \begin{gathered}A_{\triangle}=\frac12 bh\\A_{\text{trapezium}}=\frac12(a+b)h\end{gathered}\)
  2. Composite solids Add the volumes of non-overlapping components. Subtract an empty or removed prism instead
  3. Dimensions and units Convert lengths before multiplying, or convert the final cubic measure using a cubed length factor
  4. Cylinder radius and precision Halve a given diameter. Keep π exact through volume and unit conversion, then round only the final requested answer

Worked example

A block measures 9 cm by 7 cm by 5 cm. A circular hole of radius 2 cm passes straight through the 5 cm thickness. Find the remaining volume to 1 decimal place. Let B be the block volume and C the hole volume

\[V=B-C\]
  1. \(\displaystyle B=9\times7\times5=315\text{ cm}^3\)Find the solid block volume before drilling
  2. \(\displaystyle C=\pi(2)^2(5)=20\pi\text{ cm}^3\)The hole length is the full 5 cm thickness
  3. \(\displaystyle \begin{gathered}V=(315-20\pi)\text{ cm}^3\\V\approx252.2\text{ cm}^3\end{gathered}\)Subtract removed space and round once. Surface contact areas are not volume corrections

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 factor
Dropping 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 half the product of its base and perpendicular height as its area, so leaving out one-half doubles the volume
Confusing 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 volume

Difficulty levels

introRectangular-prism volume
standardFind triangular-prism, trapezoidal-prism or cylinder volume
challengeFind missing lengths and heights, or combine prism volumes
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 is applied three times 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

Why do we square the radius?+

A circle's area uses the square of its radius. Multiplying that area by the cylinder height gives its volume

Do joined solids need a contact-face correction for volume?+

No. Two non-overlapping components that meet at a face have total volume equal to the sum of their volumes. Contact faces have area, not volume. Subtract only actual overlapping or removed space

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)