GEOMETRY3D GeometryMathematics Calculator
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Surface Area — 7 Common 3D Shapes

Compute surface area and volume for sphere, cube, cylinder, cone, rectangular prism, triangular prism, and square pyramid. One calculator for all standard polyhedra and solids of revolution.

Concept Fundamentals
A = 4πr²
Sphere
A = 6a²
Cube
A = 2πr² + 2πrh
Cylinder
A = πr² + πrs
Cone

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Surface area is in square units; volume in cubic units Sphere has the smallest surface area for a given volume Lateral surface area excludes top and bottom bases Cone and pyramid formulas use slant height (computed from r, h) Use consistent units—result matches input unit squared

Key quantities
A = 4πr²
Sphere
Key relation
A = 6a²
Cube
Key relation
A = 2πr² + 2πrh
Cylinder
Key relation
A = πr² + πrs
Cone
Key relation

Ready to run the numbers?

Why: Surface area determines material needed for packaging, paint coverage, heat transfer, and chemical reactions. Different shapes require different formulas—this calculator handles seven common 3D solids.

How: Select a shape, then enter its dimensions (radius, height, side lengths, etc.). Each shape uses its standard formula: sphere 4πr², cube 6a², cylinder 2πr²+2πrh, cone πr²+πrs, and prisms sum of face areas.

Surface area is in square units; volume in cubic unitsSphere has the smallest surface area for a given volume
Sources:Wolfram MathWorld — Surface AreaKhan Academy — 3D Geometry

Run the calculator when you are ready.

Surface Area CalculatorSelect a shape and enter dimensions to compute surface area and volume
📐
3D GEOMETRYSurface Area

Surface Area — 7 Shapes

Sphere, cube, cylinder, cone, rectangular prism, triangular prism, square pyramid.

Select Shape

Sample Examples

Dimensions

surface_area.sh
CALCULATED
$ calc --shape=sphere
Surface Area
1,256.6371 cm²
Volume
4,188.7902 cm³
Share:
Surface Area
1,256.64 cm²
numbervibe.com/calculators/mathematics/3d-geometry/generic-surface-area-calculator

Visualization

rr = 10

Property Radar

Property Comparison

Property Breakdown

📐 Calculation Breakdown

INPUT
Radius
10 cm
RESULT
Surface Area
1,256.6371 cm²
A = 4\text{pi} r^{2}
RESULT
Volume
4,188.7902 cm³
V = 4\text{pi} r^{3}/3

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🏀

Basketball surface area ~1800 cm²

— Sports

🌍

Earth surface area ~510 million km²

— Geodesy

📦

Packaging design relies on surface area for material use

— Engineering

🔬

Surface-area-to-volume ratio affects heat transfer and chemistry

— Physics

📋 Key Takeaways

  • • Surface area = total area of all faces/surfaces of a 3D shape
  • • Sphere: A=4pir2A = 4\\pi r^2 | Cube: A=6a2A = 6a^2
  • • Cylinder: A=2pir2+2pirhA = 2\\pi r^2 + 2\\pi rh | Cone: base + lateral
  • • Use consistent units; result in square units

💡 Did You Know?

🏀Basketball surface area ~1800 cm²
🌍Earth surface area ~510 million km²
📦Packaging design relies on surface area for material use
🔬Surface-area-to-volume ratio affects heat transfer and chemistry
🏗️Construction uses SA for paint, siding, roofing estimates
💊Drug dosing sometimes uses body surface area

📖 How It Works

Shape Selection

Choose from 7 common 3D shapes. Each shape has a specific formula for surface area and volume. Sphere and cube need one dimension; cylinder and cone need radius and height; prisms need base dimensions and length.

Surface Area Formulas

Surface area = total area of all faces. Sphere: 4πr². Cube: 6a². Cylinder: 2πr²+2πrh. Cone: πr²+πrs (slant height). Prisms: sum of base and lateral face areas.

Volume Formulas

Volume = space inside. Sphere: 4πr³/3. Cube: a³. Cylinder: πr²h. Cone: πr²h/3. Prisms: base area × length. Pyramid: base area × height / 3.

🎯 Expert Tips

Unit Consistency

All dimensions in same unit. Result in square units.

Lateral vs Total

Cylinder/cone: lateral = curved part; total includes bases.

Triangle Base

Triangular prism: ensure sides form valid triangle.

Slant Height

Cone/pyramid: formulas use slant height (computed from r,h).

⚖️ Shape Formulas

ShapeFormula
Sphere4πr²
Cube6a²
Cylinder2πr²+2πrh
Coneπr²+πrs
Rect. Prism2(lw+lh+wh)

❓ FAQ

Surface area vs volume?

SA = outer covering (square units). Volume = space inside (cubic units).

What is lateral surface area?

Area of side faces only, excluding top and bottom bases.

Why different formulas?

Each shape has different face configuration and geometry.

Units?

Dimensions in cm → SA in cm², V in cm³. Same for m, in, ft.

Why does cone need slant height?

Lateral area uses slant height. We compute it from r and h using Pythagoras.

Can I use this for irregular shapes?

No. These formulas apply only to standard right prisms, pyramids, and solids of revolution.

How accurate are the results?

Exact for standard shapes. Use consistent units for best accuracy.

What is the surface-area-to-volume ratio?

SA/V = surface area ÷ volume. Smaller objects have higher ratio.

📊 Stats

7
Shapes
1
Output
2-4
Inputs per shape
m²
Square units

⚠️ Disclaimer: Results for standard shapes. Use consistent units.

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