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Prisms and Cuboids

A prism has a uniform cross-section along its length. The volume is always: cross-section area × length. Cuboids are rectangular prisms — the most common shape in engineering.

All prisms: V = Across × L    SA = 2×Across + perimeter × L
Cuboid formulas
Volume: V = l × w × h
SA = 2(lw + lh + wh)
Space diagonal: d = √(l² + w² + h²)
l=8cm, w=5cm, h=3cm. Find V, SA, diagonal.
V: 8×5×3 = 120 cm³
SA: 2(40+24+15) = 2×79 = 158 cm²
Diagonal: √(64+25+9) = √98 = 9.90 cm
Triangular prism formulas
V = ½ × b × h × L (b=base, h=tri height, L=length)
SA = bL + 3sL + 2×(½bh) (if equilateral)
More generally: SA = 2×(triangle area) + perimeter×L
Right-angle triangle: legs 6,8cm, L=15cm.
Hypotenuse: √(36+64) = 10 cm
V: ½×6×8×15 = 360 cm³
SA: 2×(½×6×8) + (6+8+10)×15 = 48+360 = 408 cm²
💡 Any prism: identify the cross-section shape first, find its area and perimeter, then multiply by the length. Works for hexagonal, T-section, I-section, L-section prisms too.
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Cylinders and Cones

The cylinder is a circular prism. The cone holds exactly one-third the volume of a cylinder with the same base and height — a fact you can verify by filling one with water into the other.

Cylinder
V = πr²h
Curved SA = 2πrh (the tube)
Total SA = 2πr² + 2πrh = 2πr(r+h)
r=6cm, h=20cm. Closed cylinder.
V: π×36×20 = 720π = 2262 cm³
Curved SA: 2π×6×20 = 240π = 753.9 cm²
Total SA: 2π×6×(6+20) = 312π = 980.2 cm²
Cone
V = ⅓πr²h (= ⅓ × cylinder)
Slant height: l = √(r² + h²)
Curved SA = πrl
Total SA = πr² + πrl = πr(r+l)
r=5cm, h=12cm. Find V, l, SA.
Slant: l = √(25+144) = √169 = 13 cm
V: ⅓π×25×12 = 100π = 314.2 cm³
Total SA: π×5×(5+13) = 90π = 282.7 cm²
⚠️ Slant height l ≠ h. Always use l=√(r²+h²) for surface area. h is the vertical height, l is the length along the sloping surface.
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Spheres and Frustums

The sphere is the most compact shape — smallest surface area for a given volume. The frustum is a cone with its top sliced off, common in engineering components like hoppers and reducer fittings.

Sphere
V = 4πr³/3
SA = 4πr²
Note: SA = 4 × (area of great circle)
Diameter = 18cm. Find V and SA.
r: d/2 = 9 cm
V: 4π×729/3 = 972π = 3053 cm³
SA: 4π×81 = 324π = 1018 cm²
Frustum
V = πh(R² + Rr + r²) / 3
Slant: l = √(h² + (R−r)²)
SA = π(R+r)l + πR² + πr²
R=10cm, r=6cm, h=8cm.
Slant: l=√(64+(10-6)²)=√(64+16)=√80=8.94cm
V: π×8×(100+60+36)/3=π×8×196/3=1641 cm³
SA: π(16)×8.94+π×100+π×36=970 cm²
🎛️

Interactive Shape Explorer

Choose a shape, adjust its dimensions, and see Volume and Surface Area update live.

ℹ️ How to use: Click a shape button, then drag the sliders to change its dimensions. Volume, surface area and other values update live and the diagram redraws instantly.

🎛️ Pick a shape and drag the sliders

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Volume
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Surface Area
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Change the sliders to explore how volume and surface area scale with dimension.
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Composite Shapes

Real engineering components are rarely simple. A composite shape is made of two or more basic shapes combined or subtracted. Split, calculate each, then combine.

Method: Split into basic shapes → find V and SA of each → add volumes, adjust SA for any hidden faces
Cylinder with hemispherical end

A pressure vessel: cylinder r=8cm, h=30cm, with a hemispherical cap on one end.

Find total volume and outer surface area
Cylinder V: π×64×30 = 1920π cm³
Hemisphere V: ½×4π×512/3 = 1024π/3 cm³
Total V: 1920π+1024π/3 = 7083π/3 = 7424 cm³
Cylinder curved: 2π×8×30 = 480π cm²
One flat end: π×64 = 64π cm²
Hemisphere SA: 2π×64 = 128π cm²
Total SA: (480+64+128)π = 672π = 2111 cm²
💡 The circle where cylinder meets hemisphere is NOT included in the surface area — it's an internal join.
Cylinder with cone on top

A grain silo: cylinder r=4m, h=12m, with a conical roof r=4m, h=3m.

Total volume and outer surface area
Cylinder V: π×16×12 = 192π m³
Cone V: ⅓π×16×3 = 16π m³
Total V: 208π = 653.5 m³
Cone slant: l=√(16+9)=√25=5m
Cone curved SA: π×4×5=20π m²
Cylinder curved: 2π×4×12=96π m²
Base disc: π×16=16π m²
Total SA: (20+96+16)π = 132π = 414.7 m²
Hollow cylinder (annular prism / pipe)

Pipe: outer radius R=60mm, inner radius r=50mm (wall=10mm), length=2m=2000mm.

Volume of material (wall)
Outer V: π×3600×2000 = 7,200,000π mm³
Inner V: π×2500×2000 = 5,000,000π mm³
Wall V: 2,200,000π = 6,912,000 mm³ = 6912 cm³
Surface area (all surfaces)
Outer curved: 2π×60×2000 = 240,000π mm²
Inner curved: 2π×50×2000 = 200,000π mm²
Two annular ends: 2×π(3600-2500) = 2200π mm²
Total SA: 442,200π = 1,389,300 mm²
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Applications

Where this topic is used in engineering, manufacturing, maintenance and daily life.

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Engineering Applications of 3D Shapes

Every branch of T Level / Level 3 Engineering uses volume and surface area calculations. These are the most examined contexts.

🛢️ Storage Tank Design — Cylindrical Vessel

Oil storage tanks are cylinders. Volume determines capacity; surface area determines material and insulation cost.

Tank: D=3.6m (r=1.8m), height=5.5m

V = πr²h = π×3.24×5.5 = 17.82π = 55.97 m³
= 55,970 litres

Curved SA = 2π×1.8×5.5 = 19.8π = 62.2 m²
Two ends = 2×π×3.24 = 6.48π = 20.4 m²
Total SA = 82.6 m²

Steel plate 6mm at 47.1kg/m²:
Mass = 82.6×47.1 = 3890 kg

🔩 Turned Component — Volume of Material

A stepped shaft has different diameters along its length. Total material volume = sum of cylinder volumes for each step.

Shaft: 3 sections
Section 1: d=80mm, L=120mm
Section 2: d=60mm, L=80mm
Section 3: d=40mm, L=60mm

V₁ = π×40²×120 = 192,000π mm³
V₂ = π×30²×80 = 72,000π mm³
V₃ = π×20²×60 = 24,000π mm³
Total = 288,000π = 904,779 mm³ ≈ 905 cm³

Steel density 7.85g/cm³:
Mass = 905×7.85 = 7.1 kg

🏗️ Concrete Foundation — Volume and Cost

A strip foundation for a building is a long rectangular prism. Pad foundations are cuboids or cylinders.

Strip foundation: 500mm wide, 300mm deep
Total length = 48m

Cross-section A = 0.5×0.3 = 0.15 m²
V = 0.15×48 = 7.2 m³

Pad foundation (circular): r=0.6m, h=0.5m
V = π×0.36×0.5 = 0.18π = 0.565 m³

Ready-mix concrete at £140/m³:
Strip total = 7.2×140 = £1,008

⚙️ Sheet Metal Fabrication — Unfolded Surface Area

A cylindrical duct is cut from sheet metal. The flat sheet area equals the curved surface area — useful for material costing.

Duct: diameter=300mm, length=2.5m
r=150mm=0.15m

Curved SA = 2πr×L = 2π×0.15×2.5
= 0.75π = 2.356 m²

Sheet metal at 7.5kg/m² (1mm steel):
Mass = 2.356×7.5 = 17.7kg per length

For a cone reducer (R=200mm→r=120mm, h=300mm):
Slant l = √(300²+80²) = 310 mm
Curved SA = π(R+r)l = π×320×310 = 99,200π mm²
= 0.312 m²

🌡️ Insulation — Surface Area for Heat Loss

Heat loss through a surface is proportional to area. Engineers calculate total surface area of a vessel to size insulation.

Pressure vessel: cylinder r=0.5m, h=2m
+ two hemispherical ends

Cylinder curved: 2π×0.5×2 = 2π m²
Two hemispheres = 4πr² = 4π×0.25 = π m²
Total SA = 3π = 9.42 m²

Heat loss: Q = U×A×ΔT
U=0.5 W/m²K, ΔT=60°C:
Q = 0.5×9.42×60 = 282.6 W

With 50mm insulation (U=0.08):
Q = 0.08×9.42×60 = 45.2 W (84% reduction)

🪣 Hopper — Frustum Volume and Flow

A hopper (bin bottom) is a frustum. Its volume determines storage; its slope angle determines whether material flows freely.

Hopper: R=1.2m, r=0.15m, h=1.8m

V = πh(R²+Rr+r²)/3
= π×1.8×(1.44+0.18+0.0225)/3
= π×1.8×1.6425/3
= 0.9855π = 3.096 m³

Slant: l=√(1.8²+(1.2-0.15)²)=√(3.24+1.1025)
=√4.3425=2.084m

Curved SA: π(1.2+0.15)×2.084=π×1.35×2.084
= 2.813π = 8.84 m²
🧮

3D Shape Calculator

Choose any 3D shape, enter its dimensions, and get volume and surface area with every formula and step shown clearly.

📦 Volume & Surface Area Calculator8 shapes with full working

ℹ️ How to use: Use the dropdown to pick a 3D shape. The input boxes update to match that shape. Type the dimensions and step-by-step working appears instantly.

Select a shape and enter the dimensions above.
👆 Try: Cube s=4 · Cylinder r=3, h=10 · Cone r=4, h=9 · Sphere r=6 · Frustum R=5, r=3, h=8
Volume formulas
Cube: V = s³
Cuboid: V = l × w × h
Cylinder: V = πr²h
Cone: V = ⅓πr²h
Sphere: V = &frac43;πr³
Prism: V = cross-section area × length
Surface area formulas
Cube: SA = 6s²
Cuboid: SA = 2(lw + lh + wh)
Cylinder: SA = 2πr² + 2πrh
Cone: SA = πr² + πrl  (l = slant height)
Sphere: SA = 4πr²
💡 Slant height l = √(r² + h²)
🎮

Quick Fire Quiz

Test your knowledge — 10 questions, instant feedback.

⚡ 3D Shapes Blitz

ℹ️ How to play: A 3D shapes question appears. Click the correct answer. Score and streak update after each — click Next to continue.

Score: 0 Streak: 0 🔥 Q 1/10
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✏️

Practice Questions

Always state the formula first, then substitute. Give exact answers in π where appropriate, then convert to decimal.

🟢 Tier 1 — Basic Shapes

ℹ️ How to use: Work through each question and write down your answer. When ready, click Show Answer for the full worked solution.

Q1
A cube has edge length 7cm. Find its volume and surface area.
[4]
V = 7³ = 343 cm³
SA = 6×49 = 294 cm²
Q2
A cuboid measures 12cm × 8cm × 5cm. Find volume, surface area, and the longest diagonal.
[5]
V = 12×8×5 = 480 cm³
SA = 2(96+60+40) = 2×196 = 392 cm²
d = √(144+64+25) = √233 = 15.26 cm
Q3
A cylinder has r=4.5cm and h=14cm. Find its volume and total surface area.
[4]
V = π×20.25×14 = 283.5π = 890.6 cm³
SA = 2π×4.5×(4.5+14) = 9π×18.5 = 166.5π = 523.1 cm²
Q4
A cone has base diameter 16cm and vertical height 15cm. Find its volume, slant height and total surface area.
[5]
r=8cm
l=√(64+225)=√289=17cm
V=⅓π×64×15=320π=1005 cm³
SA=π×8×(8+17)=200π=628.3 cm²
Q5
A sphere has surface area 200π cm². Find its radius and volume.
[4]
SA=4πr²=200π → r²=50 → r=?
4πr²=200π → r²=50 → r=√50=7.07cm
V=4π×50√50/3=200√50π/3=1481 cm³
🟡 Tier 2 — Engineering Problems
Q6
A steel tank is a cylinder, D=2.0m, h=4.5m. Find (a) volume in litres, (b) mass of steel if the plate is 8mm thick and density 7850 kg/m³.
[6]
(a) r=1m: V=π×1×4.5=4.5π=14.14 m³=14,140 litres
(b) SA=2π×1×(1+4.5)=11π=34.56m²
Wall thickness=0.008m → volume of steel=34.56×0.008=0.277m³
Mass=0.277×7850=2174 kg
Q7
A concrete pile (cylinder) has d=400mm and length 8m. Find volume and mass (concrete density 2400 kg/m³).
[4]
r=0.2m: V=π×0.04×8=0.32π=1.005 m³
Mass=1.005×2400=2412 kg
Q8
A hopper (frustum) has R=0.8m, r=0.1m, h=1.2m. Find volume and slant surface area.
[5]
V=π×1.2×(0.64+0.08+0.01)/3=π×1.2×0.73/3=0.292π=0.917 m³
l=√(1.44+(0.8-0.1)²)=√(1.44+0.49)=√1.93=1.389m
Curved SA=π(0.8+0.1)×1.389=π×0.9×1.389=1.25π=3.928 m²
🔴 Tier 3 — Composite & Challenge
Q9
A grain silo consists of a cylinder (r=3m, h=10m) with a hemispherical dome on top. Find total volume, and total external surface area (no floor).
[6]
Cylinder V=π×9×10=90π m³
Hemisphere V=⅔π×27=18π m³
Total V=(90+18)π=108π=339.3 m³

Cylinder curved=2π×3×10=60π m²
Hemisphere SA=2π×9=18π m²
Total SA=(60+18)π=78π=245.0 m²
Q10
A hollow steel shaft: outer d=120mm, inner d=80mm (bore), length=2.5m. Find (a) volume of steel, (b) mass at 7850 kg/m³, (c) total surface area including bore.
[7]
R=60mm=0.06m, r=40mm=0.04m
(a) V=π(R²-r²)L=π(0.0036-0.0016)×2.5=π×0.002×2.5=0.005π=0.01571 m³
(b) Mass=0.01571×7850=123.3 kg
(c) Outer curved=2π×0.06×2.5=0.3π m²
Inner curved=2π×0.04×2.5=0.2π m²
Two annular ends=2×π(0.0036-0.0016)=0.004π m²
Total=0.504π=1.583 m²
🎯 Score guide: Q1–5 foundations (22 marks) · Q6–8 engineering (15 marks) · Q9–10 challenge (13 marks)
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SkillLondon — T Level / Level 3 Engineering Maths  ·  3D Shapes: Volume & Surface Area