Geometric Formulas for Three-Dimensional Solid Shapes
Three-dimensional geometry evaluates physical solids in Euclidean space. Volume quantifies the cubic capacity contained within the closed surface, whereas Surface Area measures the combined two-dimensional boundary skin enclosing that space.
Standard 3D Solid Geometry Equations
| Solid Shape | Volume Formula (V) | Total Surface Area (A) | Lateral Surface Area (L) |
|---|---|---|---|
| Cube | s³ | 6s² | 4s² |
| Rectangular Prism | l × w × h | 2(lw + lh + wh) | 2h(l + w) |
| Sphere | (4/3) × π × r³ | 4 × π × r² | 4 × π × r² |
| Cylinder | π × r² × h | 2πrh + 2πr² | 2πrh |
| Cone | (1/3) × π × r² × h | πr(r + √(r² + h²)) | πr√(r² + h²) |
| Square Pyramid | (1/3) × a² × h | a² + 2a√((a/2)² + h²) | 2a√((a/2)² + h²) |
Frequently Asked Questions (FAQ)
What units are volume and surface area measured in?
Volume is always expressed in cubic units (e.g. cm³, in³, m³, ft³ or liters/gallons). Surface area is expressed in square units (e.g. cm², in², m², sq ft).
How does scaling dimensions affect volume and surface area?
By the square-cube law: doubling all linear dimensions of any 3D object quadruples its surface area (2² = 4x) and octuples its internal volume (2³ = 8x).