Volume of prisms: cuboids, triangular prisms and cylinders
Finding volume as cross-sectional area times length, for cuboids, triangular prisms and cylinders
About three lessons of 45 to 60 minutes
Every prism, one big idea: stack the cross-section
A cereal box, a chocolate bar with a triangular cross-section (like Toblerone), and a can of soup are all PRISMS: a flat cross-sectional shape repeated (extruded) along a length. Once you can find the area of that flat cross-section, finding the volume of the whole solid is just 1 more multiplication.
This unit builds the same big idea (volume = cross-sectional area x length) across 3 shapes: cuboids (rectangular cross-section), triangular prisms (right-angled triangle cross-section), and cylinders (circular cross-section, using pi from the previous unit).
- A cereal box: 20 cm x 8 cm x 30 cmvolume = 20 x 8 x 30 = 4,800 cm3
- A triangular-cross-section chocolate barvolume = (triangle cross-section area) x length
- A cylindrical soup can: radius 4 cm, height 12 cmvolume = pi x x 12 = 192Ο β 603.2 cm3
What students will be able to do
Students will find the volume of a cuboid and work backward from a volume to a missing dimension, find the volume of a triangular prism from its cross-sectional area and length, and find the volume of a cylinder both in terms of pi and as a rounded decimal.
- I can find the volume of a cuboid using length x width x height.
- I can work backward from a cuboid's volume and 2 known dimensions to find the missing dimension.
- I can find the volume of a triangular prism by finding the cross-sectional area first, then multiplying by the length.
- I can find the volume of a cylinder, in terms of pi and as a rounded decimal.
Standards this unit teaches
- KS3 Maths: Geometry and measuresUK National Curriculum (England)Geometry and measures
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Geometry and measures" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)" (this unit builds the volume half; a separate unit, ukYear8AreaPerimeter in lib/teachingUnits_secondary.ts, builds the perimeter/area half); "use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Prism
- a 3D solid with the same cross-sectional shape all along its length
- Cross-section
- the flat 2D shape you would see if you sliced straight through a prism, perpendicular to its length
- Cuboid
- a prism with a rectangular cross-section; a box shape with 6 rectangular faces
- Volume
- the amount of 3D space a solid takes up, measured in cubic units (e.g. cm3)
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Volume of cuboids
ConcreteA cuboid's cross-section is a rectangle, length x width, and 'stacking' that rectangle up to the height gives volume = length x width x height. Every one of the 3 dimensions plays an equal role, so the formula works whichever face you call the 'base'.
Sometimes the volume is already known (from a spec sheet, a packaging label) and 1 dimension is missing. Rearranging the formula, the missing dimension = volume / (product of the other 2), a division rather than a multiplication.
A cuboid has length 8 cm, width 5 cm and height 4 cm. Find its volume. Then a different cuboid has length 6 cm, width 3 cm and a volume of 90 cm3. Find its height.
- First cuboid: volume = length x width x height = 8 x 5 x 4 = 160 cm3.
- Second cuboid: height = volume / (length x width) = 90 / (6 x 3) = 90 / 18 = 5 cm.
Answer: The first cuboid has volume 160 cm3. The second cuboid has height 5 cm.
- A cube has a side length of 5 cm. Find its volume (a cube is a cuboid with all 3 dimensions equal).
- Why does finding a missing dimension use DIVISION instead of multiplication?
2. Volume of triangular prisms
PictorialA triangular prism's cross-section is a triangle instead of a rectangle, so the FIRST step changes: find the triangle's area (1/2 x base x height of the triangle), then multiply by the prism's length exactly as before.
It is easy to mix up the triangle's OWN height (perpendicular between its base and opposite vertex) with the prism's length (how far the triangle is extruded). Keep the 2-step order strict: cross-sectional area first, then x length.
A triangular prism has a right-angled triangle cross-section with legs of 6 cm and 8 cm, and the prism is 10 cm long. Find its volume.
- Cross-sectional area = 1/2 x 6 x 8 = 24 cm2.
- Volume = cross-sectional area x length = 24 x 10 = 240 cm3.
Answer: 240 cm3.
- A triangular prism has a right-angled triangle cross-section with legs of 5 cm and 4 cm, and is 12 cm long. Find its volume.
- Why must the cross-sectional area be found BEFORE multiplying by the prism's length, not the other way round?
3. Volume of cylinders
AbstractA cylinder is a prism whose cross-section is a circle, so volume = (circle area) x length = pi x x length. Everything from the previous unit's circle-area work carries over directly, just with 1 more multiplication for the length.
As with circle area and circumference, give the exact form (a whole number times pi) first, then a rounded decimal from the real value of pi, never a hand-rounded '3.14' used too early in the working.
A cylindrical can has a radius of 3 cm and a height of 10 cm. Find its volume, in terms of pi and rounded to 1 decimal place.
- Volume = pi x x length = pi x x 10 = pi x 9 x 10 = 90Ο cm3 exactly.
- As a decimal: 90 x 3.14159... β 282.7 cm3.
Answer: 90Ο cm3 β 282.7 cm3.
- A cylinder has a diameter of 8 cm and a length of 5 cm. Find its volume in terms of pi.
- Which of the cylinder's 2 given measurements, radius or length, gets SQUARED in the volume formula, and why only that one?
Common misconceptions and how to address them
MisconceptionVolume is found by adding length, width and height instead of multiplying them.
Why it happens: Students confuse volume with perimeter-style reasoning, where lengths are added around a boundary.
How to address it: Connect back to the stacked-layers picture: the base layer has length x width squares, and there are height many layers stacked up, so the total count is a MULTIPLICATION (repeated addition of the same layer), not a single addition of 3 lengths.
MisconceptionFor a triangular prism, multiply all 3 numbers (both legs and the length) straight away.
Why it happens: Students see 3 numbers in the question and multiply them all together, skipping the halving step the triangle-area formula requires.
How to address it: Insist on the 2-step order every time: FIRST find the cross-sectional area (which already includes the 1/2 for a triangle), THEN multiply that single area by the length. Never multiply all 3 raw numbers together in 1 step.
MisconceptionA cylinder's volume doubles when its RADIUS doubles.
Why it happens: Students expect every dimension to scale volume by the same factor, missing that radius is SQUARED in the formula while length is not.
How to address it: Show with numbers: doubling the length only doubles the volume, but doubling the radius affects BOTH the r in pi x r x r, so the volume actually quadruples. Squared measurements scale differently from linear ones.
Guided practice (with answers)
1. A cuboid has length 4 cm, width 3 cm and height 6 cm. Find its volume.
Answer: 72 cm3, because 4 x 3 x 6 = 72.
2. A cuboid has length 5 cm, height 4 cm and a volume of 100 cm3. Find its width.
Answer: 5 cm, because 100 / (5 x 4) = 5.
3. A triangular prism has a right-angled triangle cross-section with legs of 3 cm and 4 cm, and is 8 cm long. Find its volume.
Answer: 48 cm3, because (0.5 x 3 x 4) x 8 = 6 x 8 = 48.
4. A cylinder has a radius of 2 cm and a length of 9 cm. Find its volume in terms of pi.
Answer: 36Ο cm3, because pi x x 9 = pi x 4 x 9 = 36Ο.
Independent practice worksheets
Practise volume with computed, never-wrong answer keys.
Differentiation
- Build a physical stack of unit cubes (or a picture of one) to make length x width x height concrete before moving to abstract numbers.
- Provide a written 2-step checklist for prisms: (1) find the cross-sectional area, (2) multiply by the length. Tick each step off.
- Keep a circle-area formula card visible during the cylinder section, since it reuses that exact skill.
- Practise cuboid volume with all 3 dimensions given before introducing the 'missing dimension' reverse problems.
- Investigate how volume changes when every dimension of a cuboid is doubled (it multiplies by 8, not 2), and explain why using the formula.
- Find the volume of a composite 3D solid built from a cuboid and a triangular prism joined together.
- Research the surface area of a cylinder (2 circles plus a curved rectangle when unrolled) as an extension beyond this unit's volume focus.
- Compare 2 different cylinders with the same volume but different radius-to-length ratios, and discuss which is more efficient to manufacture or ship.
Assessment: exit ticket
A three-question exit ticket sampling cuboid volume, triangular-prism volume, and cylinder volume.
1. A cuboid has length 7 cm, width 4 cm and height 3 cm. Find its volume.
Answer: 84 cm3, because 7 x 4 x 3 = 84.
2. A triangular prism has a right-angled triangle cross-section with legs of 4 cm and 5 cm, and is 6 cm long. Find its volume.
Answer: 60 cm3, because (0.5 x 4 x 5) x 6 = 10 x 6 = 60.
3. A cylinder has a radius of 5 cm and a length of 4 cm. Find its volume in terms of pi.
Answer: 100Ο cm3, because pi x x 4 = pi x 25 x 4 = 100Ο.
Teacher notes and timings
- Rough timing across 3 lessons: Lesson 1 cuboids (section 1), Lesson 2 triangular prisms (section 2), Lesson 3 cylinders plus the exit ticket (section 3).
- This unit deliberately reuses the SAME 'cross-sectional area x length' idea across all 3 shapes, so lead with that single big idea before the shape-specific formulas, rather than teaching 3 unconnected formulas.
- Every cylinder answer is given both exactly (a whole number times pi) and as a genuine decimal from the real Math.PI constant, matching the previous unit's circle-area convention and this site's 'never wrong' answer-key policy.
- Prerequisite check: section 2 assumes triangle area (1/2 x base x height) and section 3 assumes circle area (pi x ) are already secure from ukYear8AreaPerimeter; revisit that unit first if either foundation is shaky.
- Present and print both work: build the base-layer array figure live on the board for cuboids, then print the 3 linked worksheets for independent practice.