Skip to content
MathAnvil
§ Geometry

3D Formulas (Volume & Surface Area)

§ Geometry

3D Formulas (Volume & Surface Area)

CCSS.6.GCCSS.8.G3 min read

Three-dimensional formulas calculate the volume and surface area of solid shapes like cubes, spheres, cylinders, and cones. Volume measures the space inside a shape in cubic units, whilst surface area measures the total area of all outer faces in square units. These formulas form the foundation of spatial mathematics at GCSE level and beyond.

§ 01

Why it matters

3D formulas appear throughout engineering, architecture, and manufacturing. A packaging designer calculating how much cardboard wraps around a cylindrical tin uses surface area formulas, whilst determining how much soup fits inside requires volume calculations. Construction workers use cuboid volume formulas to order concrete for foundations measuring 12m × 8m × 0.3m, calculating 28.8 cubic metres needed. Pharmaceutical companies rely on sphere volume formulas to determine medication dosages in spherical pills. Water companies use cylinder volume calculations to size storage tanks holding 50,000 litres. These calculations underpin GCSE mathematics and A-level further mathematics, appearing in mechanics modules where students calculate the mass of 3D objects by multiplying volume by density.

§ 02

How to solve 3d formulas (volume & surface area)

3D Surface Area & Volume Formulas

  • Cuboid SA = 2(lw + lh + wh), V = lwh.
  • Cylinder SA = 2πr² + 2πrh, V = πr²h.
  • Cone SA = πr² + πrl, V = ⅓πr²h.
  • Sphere SA = 4πr², V = ⁴⁄₃πr³.

Example: Cylinder r=3, h=10: V = π(9)(10) ≈ 282.7.

§ 03

Worked examples

Beginner§ 01

What is the volume of a cube with side 7 cm?

Answer: 343 cm³

  1. Apply formula: V = s³ → V = 7³ = 343 cm³ — Volume of a cube = side³ = 7³ = 343 cm³.
Easy§ 02

Find the surface area of a cube with side 4 cm.

Answer: 96 cm²

  1. Apply formula: SA = 6s² → SA = 6 × 4² = 6 × 16 = 96 cm² — A cube has 6 faces, each s² = 16 cm², so total = 96 cm².
Medium§ 03

Find the volume of a cuboid with length 10 cm, width 6 cm, and height 6 cm.

Answer: 360 cm³

  1. Apply formula: V = l × w × h → V = 10 × 6 × 6 = 360 cm³ — Volume = length × width × height = 10 × 6 × 6 = 360 cm³.
§ 04

Common mistakes

  • Confusing volume and surface area units, writing a cube with side 5cm has volume 125cm² instead of 125cm³
  • Forgetting to square the radius in cylinder volume, calculating V = π × 3 × 10 = 94.2 instead of V = π × 9 × 10 = 282.7
  • Using diameter instead of radius in sphere formulas, calculating surface area as 4π × 6² = 452.4 instead of 4π × 3² = 113.1 when diameter is 6cm
§ 05

Frequently asked questions

What's the difference between volume and surface area?
Volume measures space inside a 3D shape in cubic units (cm³), whilst surface area measures the total area of all outer surfaces in square units (cm²). A cube with side 4cm has volume 64cm³ and surface area 96cm².
How do you remember which formula to use for cylinders?
Cylinder volume uses πr²h (like stacking circular slices), whilst surface area adds two circular ends (2πr²) plus the curved side (2πrh). For radius 3cm and height 8cm: volume = 72π cm³, surface area = 66π cm².
Why do sphere formulas have fractions?
Sphere formulas contain 4/3 and fractions because they derive from integration calculus. The volume formula ⁴⁄₃πr³ comes from rotating a semicircle, whilst surface area 4πr² equals four times the area of a great circle.
How do you calculate surface area for shapes with curved surfaces?
Curved surfaces like cylinders and cones require specific formulas. A cylinder's curved surface equals 2πrh, added to flat circular ends. Cones use πrl for the curved surface, where l is slant height, plus πr² for the base.
What happens if measurements are in different units?
Convert all measurements to the same unit first. If length is 2m, width 150cm, and height 80cm, convert to: 200cm × 150cm × 80cm = 2,400,000cm³. Mixed units give incorrect answers and wrong unit labels.
§ 06

See also

VolumeAreaRadiusDiameterBase
§ 06

Related topics

Share this article