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Surface area and volume of prisms - Mathematics Year 8

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Introduction

How much space is inside, and how much material covers the outside? Those questions need different measures, different units and different checks.

Calculate surface area and volume for three fully specified right prisms. Build the volume from a cross-section and perpendicular length, then account for every face in the surface. The written tasks include nets, an unfamiliar L-shaped cross-section and conversion errors.

Distinguish volume, surface area and units

A prism has a constant cross-section between matching parallel end faces. Moving a cross-section along the prism preserves its shape and area. In these tasks the prism is right: its connecting edges are perpendicular to the end faces. The perpendicular distance between the ends is called length ℓ here. It must not be confused with a sloping edge in a perspective picture.

Volume measures the three-dimensional space occupied. If the cross-sectional area is A, $V = A ℓ$ . Square centimetres times centimetres gives cubic centimetres. A unit layer of area A extended through ℓ units accounts for this product; the formula does not require a circular or rectangular cross-section.

Original source volume case. A triangular cross-section is given an area of 12 cm² and the perpendicular prism length is 8 cm. Its volume is 12×8=96 cm³. No extra factor of one half is needed: the given 12 already is the triangle's area. If instead base and perpendicular height were supplied, compute half their product once to obtain A before multiplying by length.

Cuboid case. A cuboid with side lengths 4 cm, 3 cm and 2 cm has volume 4×3×2=24 cm³. Choose the 4-by -3 end: A=12 cm² and ℓ=2 cm, giving 24. A different end has area 4×2=8 and perpendicular length 3, also 24. This independent choice of cross-section checks that the roles have been matched correctly.

Surface area measures the total area of the outside faces. The cuboid has two 4-by -3 faces, two 4-by -2 faces and two 3-by -2 faces. Their areas are 12, 12, 8, 8, 6, 6 cm², total 52 cm². Pairing equal faces gives the same result as the compact expression 2(12+8+6). The volume 24 and surface 52 answer different questions, so their unequal numerical values are not a contradiction.

The quantities also behave differently under unit conversion. Since 1 cm=10 mm, 1 cm²=100 mm² and 1 cm³=1000 mm³. Therefore 52 cm²=5200 mm² while 24 cm³=24000 mm³. Multiplying both by 10 would confuse length with area or volume. Label intermediate quantities as lengths, areas or volumes so a plausible-looking number cannot hide the wrong dimension.

These are ideal geometric solids. Real packaging may require tabs, overlaps, waste or thickness allowances. Those are not silently included in the ideal external surface area. State extra assumptions separately rather than presenting a model result as a manufacturing specification.

Read nets and total every prism face

Account for every face. A net unfolds the outside into a flat arrangement. For a closed cuboid, count six faces and group them into three equal opposite pairs. Counting only the three visible faces in a perspective view gives 26 cm² for the 4-by -3-by -2 cuboid, exactly half the required 52. The first error is omission, not addition. A valid surface calculation includes hidden outside faces too.

For a closed right prism with cross-section area A and perimeter P, the two ends contribute 2 A. Each side of that cross-section extends through length ℓ to make a rectangle, so the lateral rectangles together contribute Pℓ. Thus $S = 2 A + P ℓ$ . This compact formula is a summary of all faces, not permission to use a single cross-section side as P. It is being used for right prisms here; oblique faces require different treatment.

Right triangular prism. Declare a right-triangle cross-section with legs 3 cm and 4 cm and hypotenuse 5 cm. Its area is half 3×4=6 cm² and perimeter 3+4+5=12 cm. With prism length 8 cm, volume is 6×8=48 cm³. Two triangular ends have combined area 12. The three rectangles have areas 3×8=24, 4×8=32 and 5×8=40. Total surface area 12+24+32+40=108 cm². The compact check gives 2×6+12×8=108.

The first source volume case used a different cross-section area 12 and length 8, giving 96. It must not be conflated with this 3–4–5 cross-section of area 6, even though both use length 8. Similar-looking pictures do not make different stated dimensions interchangeable.

Follow Corbettmaths’ original face-by-face method for another triangular prism. Its visible 5–12–13 cross-section and length 20 differ from our 3–4–5 cross-section and length 8. Predict which two ends and three rectangles must be counted; the written course also derives volume. Original: corbettmaths, Surface Area of Prisms - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Compare the two routes: a face list is easier to audit for a new solid; 2 A+Pℓ is efficient once the complete perimeter is known. If the top is deliberately open, subtract only the specified missing face. Do not mix an open-box task with this closed-prism calculation, and do not subtract a triangular end twice because its area already used a half factor.

Compare three solids and solve an L-shaped section

Three-solid challenge. All lengths are in centimetres, areas in cm² and volumes in cm³. Each prism is closed and right. First calculate A and the entire cross-section perimeter P, then independently total its faces to check the surface formula.

Solid	Declared cross-section	A in cm²	P in cm	Length in cm	V in cm³	S in cm²
A: cuboid	Rectangle 4×3	12	14	2	24	52
B: triangular prism	Right legs 3, 4; hypotenuse 5	6	12	8	48	108
C: L-shaped prism	6×4 rectangle minus 2×2 corner	20	20	5	100	140

Define the L shape precisely. Its ordered boundary vertices are (0, 0), (6, 0), (6, 2), (4, 2), (4, 4), (0, 4), returning to (0, 0), with coordinate distances in cm. This is the 6-by -4 rectangle with the top-right 2-by -2 corner removed. Area is 24−4=20 cm². Its six boundary lengths are 6, 2, 2, 2, 4, 4, giving perimeter 20 cm. The new inward edges replace two removed outside segments; subtracting a corner does not mean subtracting all four of its edges from the perimeter.

Extend this L-shaped section perpendicularly through 5 cm. Volume 20×5=100 cm³. The two ends contribute 40 cm². Six rectangular side faces contribute 30, 10, 10, 10, 20, 20, total 100. Therefore the complete surface is 140 cm². A separate volume decomposition gives a 6-by -4-by -5 block of 120 minus a 2-by -2-by -5 block of 20, again 100 cm³. For the surface, simply subtracting the removed block's whole surface would be wrong because the cut exposes new outside faces.

Unfamiliar transfer. Keep the 3–4–5 cross-section but use length 10 cm instead of 8. A remains 6 and P remains 12, so V=60 cm³ and S=12+120=132 cm². The added 2 cm contributes 12 cm³ of volume and 24 cm² of lateral surface, while the two end areas stay unchanged. This difference check is independent of a full recalculation.

The source 12 cm² by 8 cm volume also checks by inverse calculation: 96÷8=12 cm². In a new conversion, 48 cm³ becomes 48000 mm³, whereas 108 cm² becomes 10800 mm². A face area is not a side length. A smaller printed net does not change the exact dimensions supplied in the task.

Retain all three volumes and surfaces, the full face inventory, at least one alternate cross-section or decomposition check and a corrected error. Online reasoning about nets does not demonstrate that you have physically cut, folded or measured a model accurately. Record that practical work separately if it is actually done.

Interactive Tasks

Quiz: Test Your Knowledge

Cross-section 12 cm² and length 8 cm: volume? (96 cm³) (!48 cm³) (!20 cm³) (!96 cm²)

Cuboid 4 by 3 by 2 cm: surface area? (52 cm²) (!24 cm²) (!26 cm²) (!52 cm³)

Which quantity uses cubic units? (Volume) (!Surface area) (!Perimeter) (!Edge length)

A right 3–4–5 triangle has area? (6 cm²) (!12 cm²) (!24 cm²) (!30 cm²)

That triangle has perimeter? (12 cm) (!6 cm) (!20 cm) (!5 cm)

Length 8 triangular prism with that section: surface? (108 cm²) (!48 cm²) (!96 cm²) (!54 cm²)

L section 6 by 4 minus 2 by 2: area? (20 cm²) (!24 cm²) (!28 cm²) (!16 cm²)

The stated L section has perimeter? (20 cm) (!12 cm) (!24 cm) (!40 cm)

24 cm³ expressed in mm³? (24000 mm³) (!240 mm³) (!2400 mm³) (!24 mm³)

Closed right-prism surface formula? (Twice end area plus perimeter times length) (!End area times length only) (!Perimeter times length only) (!Twice the volume)

Memory Game

Cross-section	Constant end shape used to build the prism
Volume	Three-dimensional space occupied
Surface area	Area of every outside face
Perimeter	Complete boundary length of the cross-section
Net	Flat arrangement that folds into the outside faces
Right prism	Prism with connecting edges perpendicular to its ends

Drag and Drop

24 cm³	Volume of the 4-by -3-by -2 cuboid
52 cm²	Surface area of the 4-by -3-by -2 cuboid
48 cm³	Volume of the 3–4–5 triangular prism of length 8
108 cm²	Surface area of that triangular prism
140 cm²	Surface area of the stated L prism of length 5

Crossword Puzzle

PRISM	Solid with a constant cross-section
VOLUME	Quantity measured in cubic units
SURFACE	Outside whose total area is being counted
PERIMETER	Whole cross-section boundary length
CUBOID	Solid with three pairs of rectangular faces
RECTANGLE	Side-face shape in these right prisms

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Distinguish dimensions: Classify length, perimeter, area and volume with correct units.
Use the source volume: Explain why 12 cm² by 8 cm gives 96 cm³ without another half.
Inventory a cuboid: List all six 4-by -3-by -2 face areas and their sum.
Read the net: Match opposite rectangles without treating picture pixels as lengths.

Standard

Calculate three complete solids: Use the table and retain A, P, V, S with units.
Check the triangular prism: Compare individual five faces with 2 A+Pℓ.
Audit the L boundary: Use the six ordered vertices to derive area 20 and perimeter 20.
Correct a hidden-face error: Explain why the visible-face total 26 is not the cuboid surface 52.

Advanced

Transfer to length 10: Keep the triangular cross-section and derive V 60 and S 132.
Compare decompositions: Check L volume by subtracting a smaller block; explain why surface subtraction differs.
Convert area and volume: Convert 48 cm³ and 108 cm² to millimetre units.
Explain model limits: Distinguish ideal surfaces from actual tabs, thickness, waste and observed folding work.

Learning Check

Three-solid answer key: Calculate volume and total outside surface.
Source examples kept distinct: Solve given-area volume and six-face cuboid cases.
L-shaped perimeter and faces: Use the ordered coordinates to audit the solid.
First-error and unit analysis: Correct omitted faces and linear conversion of cubic units.
Unfamiliar transfer and method choice: Extend the 3–4–5 prism to length 10 and check differences.
Portfolio and evidence: Retain full working and declare idealised assumptions.

Worked checks and assessment guidance

A: A 12, P 14, ℓ2, V 24 cm³, S 52 cm². B: A 6, P 12, ℓ8, V 48 cm³, S 108 cm². C: A 20, P 20, ℓ5, V 100 cm³, S 140 cm². All are closed right prisms.
Given triangular area 12 times length 8 gives 96 cm³, with no second half. Cuboid 4×3×2 has faces 12, 12, 8, 8, 6, 6, sum 52 cm²; its volume is 24 cm³. The separate 3–4–5 triangular section has area 6, not 12.
Boundary 6, 2, 2, 2, 4, 4 totals 20 cm. Area 24−4=20 cm². Two ends 40 plus side rectangles 30+10+10+10+20+20=100 gives surface 140 cm². Volume 120−20=100 cm³ independently.
The visible three cuboid faces sum 26 but the hidden matching three add 26.24 cm³ equals 24000 mm³ because each of three dimensions scales 10; 52 cm² equals 5200 mm². Squared/cubed factors preserve the actual quantity.
A 6 and P 12 unchanged: V 60 cm³, S 132 cm². Compared with length 8, the increases are 12 cm³ and 24 cm²; the end area remains 12 cm². Face lists aid auditing and the perimeter formula aids efficient calculation.
Keep three face inventories, volumes, surfaces, unit conversions and an independent decomposition check. Do not claim actual cutting/folding accuracy or material allowances from an ideal online geometry task.

Learning evidence

Retain complete volume and surface calculations for three right prisms, a six-edge L-section audit, independent checks and dimensional conversions. Practical construction accuracy remains separate.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, geometry, measures, statistics, fluency, reasoning and problem solving. Year 8 allocation is editorial within KS3. All numerical cases, original schematic graphs and feedback are newly authored teaching material. Actual Commons creators/licences follow; original-publisher videos supplement the complete written teaching. Reviewed 12 November 2026.

Triangular prism.svg, Triangular_prism.png: mathematics derivative work: Lovasoa (talk), CC BY-SA 3.0. Unchanged original; caption and learning activities are original.

Net of cuboid.png, A2569875, CC BY-SA 3.0. Unchanged original; caption and learning activities are original.

Cuboid.abc.png, Cuboid.png: en:User:Tomruen, CC BY-SA 4.0. Unchanged original; caption and learning activities are original.