Cube
aA solid with six congruent square faces.
Formulas: S=6a²; lateral L=4a². a is edge length.
Derivation: Four side squares plus two base squares give 4a²+2a².
Example: a=3 gives 6(3²)=54 square units.
Surface Area Calculator
Choose a shape and measurements
Advanced options
Area conversions square the length factor: 1 m = 100 cm, but 1 m² = 10,000 cm².
Use one consistent length unit. Results use the corresponding square unit.
Result and full working
Enter the required dimensions, then calculate.
The selected formula and substitution steps will appear here.
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Surface area formula reference
B is one base area, P is base perimeter, h is perpendicular height, and l is slant height. Lateral or curved area excludes bases.
| Group | Shape | Total | Lateral / curved |
|---|---|---|---|
| Prisms and Boxes | Cube | 6a² | 4a² |
| Rectangular prism | 2(lw+lh+wh) | 2h(l+w) | |
| Triangular prism | PL+2B | PL | |
| Regular prism | PL+2B | PL | |
| Cylinders and Cones | Cylinder | 2πrh+2πr² | 2πrh |
| Cone | πrl+πr² | πrl | |
| Conical frustum | π(R+r)l+πR²+πr² | π(R+r)l | |
| Capsule | 2πrh+4πr² | No base split | |
| Pyramids | Square pyramid | a²+2al | 2al |
| Rectangular pyramid | lw+l·l₁+w·l₂ | l·l₁+w·l₂ | |
| Square pyramid frustum | a²+b²+2(a+b)l | 2(a+b)l | |
| Round Solids | Sphere | 4πr² | No base split |
| Hemisphere | 3πr² | 2πr² | |
| Spherical cap | 2πRh+π(2Rh−h²) | 2πRh | |
| Ellipsoid | ≈4π((aᵖbᵖ+aᵖcᵖ+bᵖcᵖ)/3)¹/ᵖ | No base split |
Shape formulas, derivations, diagrams, and examples
All prisms, cylinders, cones, and pyramids are right solids unless stated otherwise.
Rectangular prism
l,w,hA box whose opposite rectangular faces are congruent.
Formulas: total 2(lw+lh+wh); lateral 2h(l+w).
Derivation: Add two lw bases, two lh faces, and two wh faces.
Example: l=4,w=3,h=2 gives 2(12+8+6)=52.
Triangular prism
a,b,c; length LA right prism with two congruent triangular ends.
Formulas: total PL+2B; lateral PL.
Derivation: The three rectangles combine to a P-by-L strip; add two triangles.
Example: A 3–4–5 base with L=10 gives 120+12=132.
Regular prism
n sides of length s; prism length LA right prism with congruent regular-polygon bases.
Formulas: S=PL+2B, where P=ns and B=ns²/[4tan(π/n)].
Derivation: Unwrap the lateral rectangles into one P-by-L rectangle; add two bases.
Example: A square prism with s=2,L=5 gives 40+8=48.
Right circular cylinder
r; height hA right circular cylinder has two parallel circular bases.
Formulas: total 2πrh+2πr²; curved 2πrh.
Derivation: The curved surface unwraps to a 2πr-by-h rectangle; add two circles.
Example: r=4,h=10 gives 112π≈351.86.
Right circular cone
r; heights h,lA right cone has its apex above the base center.
Formulas: total πrl+πr²; curved πrl; l=√(r²+h²).
Derivation: A circular sector forms the curved face; add one circular base.
Example: r=3,h=4 gives l=5 and total 24π≈75.40.
Conical frustum
R,r; height h or lA right cone cut by a plane parallel to its base.
Formulas: total π(R+r)l+πR²+πr²; curved π(R+r)l.
Derivation: Add the curved annular sector and the exposed circular ends.
Example: R=4,r=2,l=5 gives 50π≈157.08.
Capsule
r; straight length hA cylinder capped by two hemispheres of the same radius.
Formula: S=2πrh+4πr².
Derivation: The two hemispheres make one sphere; add its area to the cylindrical band.
Example: r=2,h=6 gives 40π≈125.66.
Square pyramid
a; height h or lA right pyramid with a square base and centered apex.
Formulas: total a²+2al; lateral 2al.
Derivation: Four triangles each have area al/2; add the square base.
Example: a=6,h=4 gives l=5 and total 96.
Rectangular pyramid
l,w; height h or slants l₁,l₂A centered right pyramid with a rectangular base.
Formulas: total lw+l·l₁+w·l₂; lateral excludes lw.
Derivation: Each pair of triangular faces combines to base edge times matching slant height.
Example: l=6,w=4,h=3 uses slants √13 and √18.
Square pyramid frustum
a,b; height h or lA square pyramid cut parallel to its base.
Formulas: total a²+b²+2(a+b)l; lateral 2(a+b)l.
Derivation: Four congruent trapezoids contribute 2(a+b)l; add both squares.
Example: a=6,b=2,l=5 gives 120.
Sphere
r or diameter dEvery point on a sphere is the same distance from its center.
Formula: S=4πr². A sphere has neither bases nor a lateral split.
Derivation: The standard spherical formula covers its one continuous curved surface.
Example: d=12 gives r=6 and 144π≈452.39.
Hemisphere
r or diameter dHalf a sphere cut through its center.
Formulas: curved 2πr²; total with base 3πr².
Derivation: Half the sphere is 2πr²; its flat cut circle adds πr².
Example: r=3 gives total 27π≈84.82.
Spherical cap
R; cap height hThe part of a sphere above a cutting plane.
Formulas: curved 2πRh; base π(2Rh−h²).
Derivation: Use the curved cap formula and the base relation a²=2Rh−h².
Example: R=5,h=2 gives total 36π≈113.10.
Ellipsoid
a,b,cA stretched sphere with three perpendicular semi-axes.
Approximation: 4π((aᵖbᵖ+aᵖcᵖ+bᵖcᵖ)/3)¹/ᵖ, where p=1.6075.
Assumption: A general ellipsoid has no elementary exact surface-area formula, so this result is labeled approximate.
Example: a=3,b=2,c=1 gives approximately 48.97.
Worked surface area examples
Total surface area of a cylinder
For r=4 cm and h=10 cm: S=2π(4)(10)+2π(4²)=80π+32π=112π cm²≈351.86 cm².
Lateral area of a cone
For r=3 m and h=4 m, l=√(3²+4²)=5 m. Then L=π(3)(5)=15π m²≈47.12 m².
Rectangular-prism surface area
For l=8 cm, w=5 cm, and h=3 cm: S=2(40+24+15)=158 cm².
Sphere area from diameter
For d=12 in, r=6 in, so S=4π(6²)=144π in²≈452.39 in².
Find a cube side from known area
If S=216 ft², then a=√(S/6)=√(216/6)=√36=6 ft.
Methodology, sources, and accuracy
Authorship
Prepared by Starlight Robotics and published by Starlight Tools. No individual credentialed mathematical reviewer is currently identified.
Prepared by Starlight Robotics and published by Starlight Tools. No individual credentialed mathematical reviewer is currently identified.
Publication and review record
A reliable original publication date and independent last-reviewed date are not documented, so none is claimed in the metadata.
A reliable original publication date and independent last-reviewed date are not documented, so none is claimed in the metadata.
Formula sources
Right-solid decomposition follows OpenStax: Volume and Surface Area. Square units and conversions follow NIST SI Units — Area.
Right-solid decomposition follows OpenStax: Volume and Surface Area. Square units and conversions follow NIST SI Units — Area.
Accuracy
Unrounded inputs are used throughout. Derived slant heights are rounded only for display. Exact π remains symbolic; the decimal uses the chosen π setting. Ellipsoid area is explicitly approximate.
Unrounded inputs are used throughout. Derived slant heights are rounded only for display. Exact π remains symbolic; the decimal uses the chosen π setting. Ellipsoid area is explicitly approximate.
Documented checks and assumptions
Representative checks cover each formula family: cube a=2→24; box 2×3×4→52; 3–4–5 triangular prism of length 10 →132; cylinder r=4,h=10→112π; cone r=3,h=4→24π total; square pyramid a=6,h=4→96; sphere r=3→36π; and hemisphere r=3→27π. Equivalent radius and diameter modes are checked against one another. Other shapes are checked by adding their displayed components.
All dimensions must be positive and use one unit. A capsule height means the straight cylinder between the ends. A spherical cap height cannot exceed the sphere diameter. A rectangular pyramid apex is above the base center. Ellipsoid area uses Knud Thomsen’s approximation with p=1.6075.
Surface Area Calculator FAQs
What is the difference between total and lateral surface area?
Total area includes every outside face. Lateral or curved area excludes bases. A closed cylinder has 2πrh+2πr²; its curved area is 2πrh.
Is surface area the same as volume?
No. Surface area measures the outside covering in square units; volume measures enclosed space in cubic units.
Why is surface area measured in square units?
Each face is two-dimensional, so two length factors are multiplied. A 3 cm by 4 cm face has area 12 cm².
Can I use diameter instead of radius?
Yes. Select a diameter mode for round solids. The calculator uses r=d/2 and shows that conversion.
Why does a cone use slant height?
The curved cone surface extends from rim to apex, so its area uses slant height l. For a right cone, l=√(r²+h²).
Does cylinder surface area include the bases?
Total cylinder area includes two circular bases. Curved area excludes both; one-base-open area includes the curved surface and one circular base.
What is curved versus total area of a hemisphere?
A hemisphere’s curved area is 2πr². Including its flat circular base gives total area 3πr².
How do I calculate an open-top object?
Choose the one-base-open or open-top mode. A cylinder uses 2πrh+πr²; a box uses its side faces plus one base.
How does scaling every dimension change surface area?
Multiplying every length by k multiplies area by k². Doubling all dimensions makes area four times as large.
How can I find a missing dimension from known surface area?
Rearrange the formula first. For a cube, a=√(S/6); for a sphere, r=√(S/(4π)).