Surface Area Calculator for Common 3D Shapes
Calculate total, open, lateral or curved surface area with the correct formula. Keep exact fractions and π forms, convert square units, and review every substitution before using the result.
Last Updated: July 26, 2026
Online Surface Area Calculator
Use one shared linear unit for every dimension. Integers, decimals, fractions, proper mixed numbers and scientific notation are supported.
Your surface-area result and formula steps will appear here.
Calculation steps
- Choose a surface mode and enter its dimensions.
Surface-area unit conversions
| Area unit | Equivalent value |
|---|---|
| Calculate a surface area with a physical input unit to see conversions. | |
Recent calculations
- Your last six results will appear here.
How to Use This Surface Area Calculator
- Choose the solid and the exact surface mode. “Total” includes every listed base or end. Open, lateral and curved-only modes exclude specific faces.
- Select one input length unit. Convert every dimension to this unit before entering values.
- Choose the output area unit and decimal precision. Generic lengths remain in generic square units.
- Enter each requested measurement. Cone, frustum and pyramid modes ask for slant height, not perpendicular height.
- Select Calculate Surface Area. Review the exact result, decimal value, formula, substitution, steps and conversion table.
Use external dimensions when you need the outside boundary. Use internal dimensions only when you are deliberately estimating an inside lining. Wall thickness, open faces, overlaps and real manufacturing details decide which surfaces belong in a practical coverage estimate.
What Surface Area Measures
Surface area measures the combined two-dimensional boundary of a three-dimensional object. A rectangular box has six faces. A closed cylinder has one curved side and two circular ends. A sphere has one continuous curved boundary. Each part contributes an area, so the final unit is squared, such as cm², m² or ft².
Surface area differs from ordinary area and volume. Area describes one flat region. Surface area adds the relevant faces of a solid. Volume measures the three-dimensional space inside or occupied by the solid. The 12 × 7.5 × 3 metre box below has surface area 297 m² and volume 270 m³, so the values and units answer different questions.
Surface Area Formulas Included
| Solid or surface | Required measurements | Formula |
|---|---|---|
| Cube | Side s | SA = 6s² |
| Rectangular prism | Length l, width w, height h | SA = 2(lw + lh + wh) |
| Triangular prism | Triangle base b, height t, perimeter P; prism length L | SA = bt + LP |
| Closed cylinder | Radius r, height h | SA = 2πr(r + h) |
| Cylinder with one open end | Radius r, height h | SA = πr² + 2πrh |
| Cylinder curved surface | Radius r, height h | LA = 2πrh |
| Hollow cylinder or tube | Outer radius R, inner radius r, height h | SA = 2πh(R + r) + 2π(R² − r²) |
| Right cone | Radius r, slant height l | SA = πr(r + l) |
| Right conical frustum | Radii R and r, slant height l | SA = π[R² + r² + (R + r)l] |
| Sphere | Radius r | SA = 4πr² |
| Hemisphere with base | Radius r | SA = 3πr² |
| Hemisphere curved only | Radius r | CSA = 2πr² |
| Right square pyramid | Base side s, face slant height l | SA = s² + 2sl |
| Capsule | Radius r, straight cylinder length c | SA = 2πrc + 4πr² |
Introductory geometry reference: OpenStax volume and surface area.
Worked Example: Rectangular Prism Surface Area
Take a closed rectangular prism with length 12 metres, width 7.5 metres and height 3 metres. It has two 12 × 7.5 faces, two 12 × 3 faces and two 7.5 × 3 faces.
SA = 2[(12 × 7.5) + (12 × 3) + (7.5 × 3)]
SA = 2(90 + 36 + 22.5)
SA = 297 m²
The answer is exactly 297 m². If the box has an open top, subtract the top face area, 90 m², to get 207 m². This shows why the selected surface scope matters as much as the dimensions.
Worked Example: Closed Cylinder Surface Area
For a closed cylinder with radius 3 metres and height 10 metres, the curved side contributes 2πrh and the two ends contribute 2πr².
SA = 2π × 3 × (3 + 10)
SA = 78π m²
SA ≈ 245.044227 m²
The exact answer remains 78π m². One-open-end mode subtracts one circle and returns 69π m². Curved-only mode excludes both circles and returns 60π m².
Total, Open, Lateral and Curved Surface Area
Total surface area normally includes every outer face of a closed solid. An open container deliberately omits one or more faces. Lateral area for a prism or cylinder means the side boundary only. Curved surface area applies the same idea to a curved object. The calculator labels each mode so a partial surface is never presented as a closed total.
For paint, fabric, insulation or sheet material, decide whether the bottom, top, inside wall, edge face or opening receives material. A formula cannot infer those project choices. Calculate separate components when different surfaces use different materials or coverage rates.
Prisms and Triangular Ends
A rectangular prism formula adds three distinct face pairs. For a triangular prism, the two triangular ends total bt because each end has area bt/2. The three rectangular side faces total LP, which is prism length multiplied by the complete triangle perimeter.
Triangular mode asks for base b, its perpendicular height t, the full end-triangle perimeter P and prism length L. A triangle exists for those base and height values only when (P − b)² is at least b² + 4t². The calculator checks this condition exactly before returning a result. Use the Triangle Calculator first when perimeter or height is unknown.
Cones, Frustums and Slant Height
Right-cone curved area is πrl. Adding the base πr² gives total area πr(r + l). Slant height l runs along the cone surface from the rim to the apex. If only perpendicular height h is known, calculate l = √(r² + h²) before using cone mode.
A right conical frustum has two circular ends and a curved band. Its curved area is π(R + r)l. Total area adds πR² and πr². Slant height must be at least |R − r|. Equality describes zero perpendicular height and is marked as a degenerate boundary.
Spheres, Hemispheres and Capsules
A sphere has surface area 4πr². A hemisphere’s curved half contributes 2πr². Adding its circular base gives total hemisphere area 3πr². Choose curved-only mode for a dome with an open circular edge, and total mode when the flat base is part of the measured boundary.
A capsule combines a cylinder with two hemispherical ends. The two ends form one complete sphere, so total area is 2πrc + 4πr². Input c is only the straight cylindrical section. When c is zero, the capsule becomes a sphere and remains a valid nondegenerate result if r is positive.
Hollow Cylinder or Tube Surface
Tube mode includes four parts: the outer curved wall, the inner curved wall and two annular end faces. It assumes concentric circles, a shared height and clean flat ends. The outer radius must be at least the inner radius. If the inner radius is zero, the formula reduces to a closed solid cylinder.
If your project needs only the inner lining, only the outside wall or an open-ended pipe without the annular rim faces, calculate the required components separately. Coatings also have thickness, waste, overlap and manufacturer coverage limits that pure geometry does not include.
Right Square Pyramid Surface Area
A right square pyramid has one square base and four congruent triangular faces. The base contributes s². The four triangles together contribute 2sl, where l is the altitude of one triangular face from the midpoint of a base edge to the apex. It is not a lateral edge.
For a centred right pyramid, face slant height cannot be less than s/2. Equality means the perpendicular height is zero, so the calculator labels the result as a degenerate boundary even when the formal face-sum is positive.
Square Unit Conversion
A linear conversion factor must be squared for surface area. Since 1 metre equals 100 centimetres, 1 m² equals 100², or 10,000 cm². Applying the factor only once gives a result 100 times too small in this example.
The calculator uses exact international length definitions. One inch is exactly 0.0254 metre, one foot is exactly 0.3048 metre, one yard is exactly 0.9144 metre and one mile is exactly 1609.344 metres. It also includes hectares and international acres for unusually large surfaces.
Conversion reference: NIST revised unit conversion factors.
Exact Fractions, π and Rounding
Every entered finite decimal, fraction, proper mixed number or scientific-notation value is parsed into a reduced rational number. Flat-faced formulas therefore stay exact within the stated limits. Curved formulas keep an exact rational coefficient multiplied by π.
Decimal displays use a 120-place π constant and convert the unrounded exact result before final rounding. The precision selector changes presentation only. Small positive values switch to scientific notation rather than appearing as zero. Real measurement accuracy still depends on your source data, so extra displayed digits do not create extra physical precision.
Zero Dimensions and Degenerate Solids
Zero is accepted because it is a useful mathematical boundary. Unlike volume, a collapsed 3D shape can retain a positive face-sum. A rectangular prism with zero height, for example, produces two coincident rectangles under the formal six-face expression. Those coincident faces are not two separately exposed physical surfaces.
The calculator therefore marks collapse conditions as degenerate even when the formula result is not zero. Use the value to study the formula boundary, not as an automatic material estimate. A capsule with zero straight length is an exception: it becomes an ordinary sphere rather than collapsing.
Measurement Assumptions and Practical Coverage
- All dimensions in one calculation use the same selected linear unit.
- Radius is half diameter. Entering diameter as radius multiplies many curved results by more than two.
- Cone, frustum and square-pyramid inputs use slant height as labeled.
- Tube mode includes inner and outer walls plus both annular end faces.
- Triangular-prism height is perpendicular to base b, and perimeter P includes all three sides of the same end triangle.
- Capsule length excludes both hemispherical ends.
- Simple formulas ignore seams, rounded corners, wall thickness, holes, fittings and surface texture unless represented in your dimensions.
For purchasing paint, cladding, fabric or insulation, apply product-specific coverage and waste after finding geometric area. Coverage may change with porosity, coats, overlap, cut layout and installation method. Safety-critical design, fabrication or engineering work requires the applicable standard and qualified review.
Common Surface Area Mistakes
- Using volume when the task asks for covering the outside.
- Choosing total area when the object is open or only its side will be covered.
- Mixing centimetres, metres, inches or feet inside one formula.
- Applying a linear conversion factor instead of squaring it.
- Entering diameter where the field asks for radius.
- Using perpendicular height in a field labeled slant height.
- Counting a shared or hidden face that is not exposed.
- Rounding dimensions, π or conversions before the final step.
Method, Limits and Result Review
Length fields accept up to 50 numeric digits, a maximum raw length of 100 characters and scientific exponents from −100 through 100. Fraction denominators must be nonzero. Proper mixed-number fractional parts must be smaller than one whole. Negative geometric lengths are rejected.
This page calculates the listed direct surface formulas. It does not solve missing dimensions, unfold printable nets, calculate arbitrary polyhedra, approximate ellipsoid area, detect overlapping faces in composite solids, subtract openings, model wall thickness, or estimate material cost. Break an irregular object into verified components and avoid double-counting shared faces.
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Frequently Asked Questions
What is surface area?
Surface area is the combined area of the faces and curved boundaries included in a three-dimensional shape. It is measured in square units.
How do you calculate total surface area?
Identify every exposed face or curved section, calculate each area with consistent units, and add the parts. The calculator applies the matching combined formula.
What is the difference between area, surface area and volume?
Area measures one flat region, surface area measures the boundary of a 3D solid, and volume measures the space inside or occupied by that solid.
Why is surface area measured in square units?
Each face or curved boundary is two-dimensional. A length conversion factor must therefore be squared, such as 1 m² = 10,000 cm².
Does this calculator accept fractions and scientific notation?
Yes. Length fields accept integers, decimals, fractions, proper mixed numbers and scientific notation within the stated input limits.
Can I enter dimensions in different units?
No. Convert every dimension to the selected input length unit before calculating. Mixing units inside one formula produces an incorrect result.
What is the difference between total and lateral or curved surface area?
Total surface area includes every stated end or base. Lateral or curved surface area excludes the bases and measures only the side boundary.
How is the total surface area of a closed cylinder calculated?
Add the curved side and two circular ends: SA = 2πrh + 2πr² = 2πr(r + h).
How is cone surface area calculated?
For a right cone, add the circular base and curved side: SA = πr² + πrl = πr(r + l), where l is slant height.
Does hemisphere total surface area include the circular base?
Yes. Total hemisphere area is 3πr², including the base. Curved-only hemisphere area is 2πr².
What happens when a dimension is zero?
Zero is accepted as a boundary. The calculator marks a collapsed solid as degenerate because coincident faces can make the formula value differ from a physical exposed surface.
How does the calculator handle pi and rounding?
Curved results keep an exact reduced coefficient times π. Decimal values use a 120-place π constant and round only at the selected final precision.