Volume of prisms and cylinders
A prism or cylinder keeps the same cross-section along its length. Find the cross-sectional area with its own formula, then multiply once by the perpendicular length
\[V=A_{\text{cross-section}}\times L\]
- Identify the cross-section Use the region that is repeated along the prism or cylinder length \(\displaystyle V=A\times L\)
- Find a triangular area A triangular cross-section uses one-half of base times perpendicular height \(\displaystyle A=\frac12\times6\times8=24\text{ cm}^2\)
- Find a circular area A circular cross-section uses the radius squared, not the diameter squared \(\displaystyle A=\pi(3)^2=9\pi\text{ cm}^2\)
- Extend the cross-section Multiply the cross-sectional area by the perpendicular length and give cubic units \(\displaystyle V=24\times10=240\text{ cm}^3\)
Worked example
A cylindrical storage vessel has radius 4 m and perpendicular height 18 m. Find its volume to 1 decimal place
\[r=4\text{ m},\quad h=18\text{ m}\]
- \(\displaystyle A=\pi(4)^2=16\pi\text{ m}^2\)Find the circular cross-sectional area by squaring the radius and multiplying by pi
- \(\displaystyle V=A\times18=16\pi\times18=288\pi\text{ m}^3\)Extend the cross-section through the vessel's perpendicular height
- \(\displaystyle 288\pi\approx904.8\text{ m}^3\)Round the exact volume once at the end to 1 decimal place
Common mistakes
Multiplying every given length
✕\(\displaystyle V=6\times8\times10\times15=7200\text{ cm}^3\)✓\(\displaystyle V=\frac12\times6\times8\times15=360\text{ cm}^3\)
The third side of the triangle is not used in its area formula, so it is not another volume factorDropping the triangular half
✕\(\displaystyle V=6\times8\times15=720\text{ cm}^3\)✓\(\displaystyle V=\frac12\times6\times8\times15=360\text{ cm}^3\)
A triangular cross-section uses one-half of the base multiplied by its perpendicular height, so leaving out one-half doubles the volumeConfusing surface area and volume
✕\(\displaystyle SA=2\left(\frac12\times6\times8\right)+(6+8+10)\times15=408\text{ cm}^2\)✓\(\displaystyle V=\frac12\times6\times8\times15=360\text{ cm}^3\)
The face sum has square units and finds surface area, while cross-sectional area times length has cubic units and finds volumeDifficulty levels
introFind the volume of a rectangular prism from its cross-sectional dimensions and length
standardFind the area of a triangular cross-section or circle before calculating the volume
challengeWork backwards, use a given cross-sectional area or combine two rectangular-prism components
examChoose the triangular-prism volume from options containing half, all-length and surface-area errors
Questions about this topic
What is a cross-sectional area?+
It is the area of the repeated face that is perpendicular to the prism length or cylinder height
Why is one-half used for a triangular prism?+
The area of a triangle is one-half of its base multiplied by its perpendicular height
Why is the answer in cubic units?+
Volume combines a two-dimensional cross-sectional area with a length, producing three dimensions
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity
In this chapter
Round to decimal places and significant figuresScientific notationMultiply and divide in scientific notationFind a side of a right-angled trianglePythagoras in three dimensions10C Perimeter and area: Convert measurement unitsPerimeter and area of standard shapesArea of a triangle from three sidesCircumference and area of a circlePerimeter and area of composite shapesArc length and sector area10E Volume and capacity: Convert measurement unitsVolume of prisms and cylinders10E Volume and capacity: Volume of cones, pyramids and spheresCapacity of a container10F Volume of pyramids: Volume of cones, pyramids and spheresSurface area of solidsSimilar figures and scale factorsTests for similar trianglesSimilar solids and scaling volume