Surface Area Calculator - Cube, Sphere, Cylinder, Cone, Prism

A surface area calculator measures the outer skin of a solid - how much wrapping, paint, or sheet metal you would need if you could unfold the object into a net. The answer is still in square units, like plane area, but you are adding faces of a 3-D body, not filling its interior. Interior fill is volume (cubic units) and lives on a different tool.

This page covers a cube (SA = 6s²), a rectangular prism (SA = 2(lw+lh+wh)), a sphere (SA = 4πr²), a closed right circular cylinder (SA = 2πr(r+h)), a right circular cone including the base (SA = πr(r+ℓ) with ℓ = √(r²+h²)), and a square pyramid (SA = a² + 2a√((a/2)²+h²)). The default is a cube of side 4: six faces of 16 give 96. That is not 64 (volume 4³) and not 16 (one face).

Work stays in the browser. For how much a tank holds, open the volume calculator. For a single rectangular or circular face, the area calculator is enough. Circle radius, diameter, and circumference belong on the circle calculator; this page only needs r when the solid is round.

Using this cube sphere cylinder cone prism surface tool takes four steps:

1. Pick the solid. A soup can with both lids on is a closed cylinder. An open soup can (no lids) would be lateral area only - this tool still reports total and shows lateral so you can subtract the bases by hand if needed.
2. Enter positive lengths in one unit. Cone and pyramid height is the perpendicular drop from apex to base, not a slanted edge.
3. Read the formula, substitution, and extras (lateral vs bases on a cylinder; slant ℓ on a cone).
4. Copy or clear. Last solid and dimensions stay locally for up to 30 days.

Note: no unit dropdown, no paint-spread rate, no wall thickness, no hollow shells. A frustum, torus, or triangular prism is not in the menu.

Worked example - cube s = 4:

Each face is 4 × 4 = 16. Six faces: 96. Volume would be 64. If you only needed one face (a square tile), that is 16 and belongs on the area page. Surface area of the die is 96.

Worked example - prism 2 × 3 × 4:

SA = 2(2·3 + 2·4 + 3·4) = 2(6 + 8 + 12) = 2 × 26 = 52. The three pair-products are the three distinct face sizes. Order of l, w, h does not matter. A cube is the special prism with l = w = h.

Packaging often lists outer l, w, h. That overstates the interior cardboard area only slightly compared with inner dimensions, but it is the right number for wrapping paper around the box. Open-top boxes omit the top lw; subtract that face yourself - this UI always includes all six prism faces.

Unroll a right circular cylinder’s side and you get a rectangle of width 2πr (the circumference) and height h, so lateral area = 2πrh. A closed can also has two disks of area πr² each. Add them: total SA = 2πr² + 2πrh = 2πr(r+h). This calculator reports the closed formula as the main number and prints the lateral piece beside it so you can match either homework wording.

Example: r = 3, h = 10. Lateral = 60π. Two bases = 18π. Total = 78π ≈ 245.04422698074857. An open cylinder (one base, like a glass) would be 2πrh + πr² = 69π; compute that by adding lateral plus one base from the extras, not by inventing a third menu item.

Height is perpendicular. An oblique cylinder with the same perpendicular h and the same base radius has the same volume (Cavalieri) but a larger lateral area; this tool does not model shear. Radius is not diameter: typing d for r overstates every term that contains r or r².

A sphere of radius r has SA = 4πr², four times the area of a great circle. That is Archimedes’ result, not a coincidence with the derivative of (4/3)πr³ with respect to r. Volume of the ball is (4/3)πr³ - different exponent, different tool. This page uses Math.PI.

Example: r = 1 gives 4π ≈ 12.566370614359172. Example: r = 2 gives 16π ≈ 50.26548245743669. Doubling r multiplies surface area by four and volume by eight. A hemisphere’s curved surface is 2πr²; including the flat disk makes 3πr². This menu is the full sphere only.

For a right circular cone, draw the right triangle with legs r (base radius) and h (perpendicular height). The hypotenuse is the slant height ℓ = √(r² + h²). Lateral area is the sector that unrolls to πrℓ. Adding the base disk πr² gives total SA = πr(r + ℓ).

Example: r = 3, h = 4. Then ℓ = 5 (a 3-4-5 triangle). Lateral = 15π. Base = 9π. Total = 24π ≈ 75.39822368615503. If you type 5 into the height box thinking it is slant, the tool will treat 5 as perpendicular h and compute a different ℓ = √(9+25) = √34, which is wrong for that problem. Height here is always the altitude.

A square pyramid with a = 6 and h = 4 has face slant √(9+16) = 5. SA = 36 + 2×6×5 = 36 + 60 = 96, coincidentally the same number as the default cube, from a completely different net. Pyramid height is the altitude to the base square, usually to the centre, not a lateral edge to a corner.

Each face is an area, so you add square units: m², cm², in². Volume would be m³. A net of the solid is a 2-D puzzle whose pieces sum to the same surface area. This calculator does not convert among unit names; enter every length in the same unit. Mixing cm with m without converting first will be wrong by 10,000 for every face that should have been metres.

Linear scale factor k changes surface area by k² and volume by k³. That is why a scale model uses much less paint than the ship, but not in the same ratio as the length scale, and why “twice as tall” is eight times the volume and four times the skin if all three dimensions scale.

• Reporting volume as surface area (64 vs 96 on the default cube).
• Using slant as cone height or forgetting to add the base disk.
• Cylinder: giving only 2πrh when the problem said “closed” or “total.”
• Sphere: writing 4πr³ or 4/3 πr². Exponents matter.
• Diameter in the radius box on sphere, cylinder, or cone.

Measure the base radius r (half the lid diameter) and the perpendicular height h of the can. Total closed surface is two lids plus the label wrap: 2πr² + 2πrh. Factor 2πr to get the compact form 2πr(r+h) that this page prints as the main result. If a homework key used 3.14, your number will sit a few thousandths away because this tool uses Math.PI.

A tin can’s “height” on a grocery label is often the outer height including the rim, while the label wrap is a bit shorter than h. Geometry class ignores that seam. An open pipe (no lids) is only 2πrh. A pipe with thickness is an outer cylinder minus an inner cylinder for volume, but for paint on the outside you still want the outer 2πR(R+H) if both rims are painted, or 2πRH if only the curved wall is painted. Read the extras line for lateral vs bases, then add the pieces that match the physical job.

A square pyramid’s net is a square of side a plus four isosceles triangles. Each triangle has base a and height equal to the slant of the face from the midpoint of a base edge up to the apex: ℓ_face = √((a/2)² + h²), where h is the perpendicular height of the pyramid, not the lateral edge to a corner. Four triangles contribute 4 × (½ a ℓ_face) = 2a ℓ_face. Adding the base gives SA = a² + 2a √((a/2)² + h²).

The lateral edge from apex to a base corner is √((a/√2)² + h²) = √(a²/2 + h²), which is longer than the face slant unless a = 0. Do not type that edge as h. Egyptian-style “rise over run” along a face is the face slant, still not h. If you only need the four triangular faces (a tent without a floor), subtract a² from the total this page reports.

• Six solids. No torus, ellipsoid, frustum, or triangular prism.
• Closed cylinder and cone include bases. Lateral is shown as extra, not as a separate mode.
• No unit conversion and no paint litres.
• IEEE-754 π. Huge r² can overflow.
• Positive lengths only.