Surface Area Tool
Find total exterior surface area for ten common solids using shape-specific dimensions. Optionally exclude selected faces, inspect each surface contribution and percentage, view illustrative diagrams, and export numerical results or the shape graphic.
| Surface | Area | Share |
|---|
Table of Contents
Highlights
- Ten supported 3D shapes, from cubes to capsules
- Optional excluded-face calculations for applicable solids
- Instant total area with surface-by-surface percentages
- Illustrated shape previews and visual contribution bars
- Export results to CSV or JSON and diagrams to SVG
How Surface Area Tool Works?
Surface area is the sum of the areas of the external faces or curved surfaces of a three-dimensional solid. The selected shape determines which dimensions are required and which mathematical expression applies. For a closed cuboid, for example:
Here l, w and h are its length, width and height. Curved solids use π and, where needed, slant height determined from the Pythagorean theorem. The following table summarises the closed-solid formulas.
| Shape | Total surface area |
|---|---|
| Cuboid | 2(lw + lh + wh) |
| Cube | 6s² |
| Sphere | 4πr² |
| Cylinder | 2πrh + 2πr² |
| Right circular cone | πr(r + √(r² + h²)) |
| Hemisphere | 3πr² |
| Right triangular prism | 2A△ + (a + b + c)L |
| Square pyramid | s² + 2s√(h² + s²/4) |
| Conical frustum | π(R + r)√((R − r)² + h²) + π(R² + r²) |
| Capsule | 2πrh + 4πr² |
For the triangular prism, A△ is the triangle area found using Heron's formula, A△ = √(p(p − a)(p − b)(p − c)), where p = (a + b + c)/2. For a capsule, h denotes only the length of its cylindrical section. The optional open-face setting removes one specified face: a cuboid top, one cube face, the cylinder top, the cone base, the hemisphere flat base, one triangular prism end, the pyramid square base or the frustum top. The result also separates surface contributions and shows each as a percentage of the total.
Practical Use Cases
Packaging designers: Estimate the external material area of closed boxes, cube packages and cylindrical containers, or exclude a top face when developing an open container.
Builders and decorators: Determine approximate covering areas for regular columns, square pyramidal features and rectangular enclosures before planning finishes.
Students and teachers: Compare formulas for spheres, cones, hemispheres, prisms and frustums, using the illustrated shape and surface-by-surface breakdown to check geometric reasoning.
Fabricators and product engineers: Estimate exposed areas of simple caps, cylindrical housings and capsule-shaped components, including their separate curved and end surfaces.
Garden and exhibition planners: Estimate the area of regular geometric displays and structures, with an option to omit a base or end that will remain uncovered.
Accuracy and Limitations
The results use standard geometric formulas and JavaScript floating-point arithmetic, with displayed values rounded to at most eight decimal places and approximately nine significant digits. Dimensions must be positive; triangular-prism sides must satisfy the triangle inequality. The formulas assume ideal shapes: a right cone, right triangular prism, vertically centred square pyramid and circular frustum; the capsule consists of a cylinder capped by two hemispheres. The preview is illustrative and not to scale. The open-face choice excludes only the specific face named beside the switch, not arbitrary surfaces, and the calculation does not account for wall thickness, seams, overlaps, cut-outs or manufacturing waste.
