3D Formulas (Volume & Surface Area)
Three-dimensional formulas calculate volume and surface area for geometric solids like cubes, cylinders, spheres, and cones. Volume measures the space inside a shape, expressed in cubic units, while surface area measures the total area of all exterior faces, expressed in square units. These formulas apply fundamental geometric principles to quantify physical properties of solid objects.
Why it matters
Architecture relies on surface area calculations to determine paint coverage and material costs — a cylindrical water tank with radius 5 feet and height 12 feet requires 534 square feet of surface coating. Manufacturing uses volume formulas to calculate container capacity and shipping costs. A spherical storage tank with radius 8 feet holds approximately 2,145 cubic feet of material. Construction projects need both measurements: concrete volume for foundations and surface area for insulation coverage. Engineers apply these formulas in fluid dynamics, where a cone-shaped funnel with radius 6 inches and height 10 inches has volume 377 cubic inches. These concepts extend into calculus applications for optimization problems and physics calculations involving density and pressure.
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.
Worked examples
What is the volume of a cube with side 6 cm?
Answer: 216 cm³
- Apply formula: V = s³ → V = 6³ = 216 cm³ — Volume of a cube = side³ = 6³ = 216 cm³.
Find the surface area of a cube with side 8 cm.
Answer: 384 cm²
- Apply formula: SA = 6s² → SA = 6 × 8² = 6 × 64 = 384 cm² — A cube has 6 faces, each s² = 64 cm², so total = 384 cm².
Find the volume of a cuboid with length 3 cm, width 4 cm, and height 8 cm.
Answer: 96 cm³
- Apply formula: V = l × w × h → V = 3 × 4 × 8 = 96 cm³ — Volume = length × width × height = 3 × 4 × 8 = 96 cm³.
Common mistakes
- Confusing surface area and volume units leads to writing cylinder surface area as 628 cubic units instead of 628 square units when radius equals 5 and height equals 10.
- Forgetting the base area in cylinder surface area gives 314 square units instead of 628 square units for a cylinder with radius 5 and height 10.
- Using diameter instead of radius in sphere formulas produces volume 14,137 cubic units instead of 1,767 cubic units when the actual radius is 7.5.
- Omitting the fraction in cone volume formula yields 942 cubic units instead of 314 cubic units for radius 5 and height 12.