Painting a cylinder or wrapping a box needs the lateral area, not the total. This gives both.
Total surface area is every face of a solid added together — how much wrapping paper it takes to cover the whole thing. Lateral surface area is only the curved or side faces, excluding the top and bottom — how much material makes just the label that wraps around a can, with no lid or base included. On this page’s own cylinder default (radius 3, height 4): lateral area is 75.40, and adding the two circular ends (2 × π × 3² = 56.55) brings the total to 131.95.
You enter the cone's vertical height, but the formula needs the slant height — the distance along the sloped surface from the tip down to the base edge, which is always longer than the vertical height. The calculator recovers it with the Pythagorean theorem, treating the radius, the vertical height, and the slant as a right triangle: slant = √(r² + h²). For radius 3 and height 4, that is √(9+16) = √25 = exactly 5 — a 3-4-5 right triangle in disguise. The lateral area then follows as πrl = π × 3 × 5 = 47.12, plus the circular base (π × 3² = 28.27) for a total of 75.40.
Slice a sphere through its centre and you get a flat circle the same radius as the sphere — a great circle. The sphere's entire surface area, 4πr², is precisely four times that one circle's area, πr². For radius 3: one great circle is 28.27, and the full sphere is 113.10 — exactly 4 × 28.27. This is not a coincidence of these particular numbers; it holds for every sphere, and it is one of the more elegant facts in solid geometry — Archimedes considered its proof (that a sphere's surface equals the curved surface of the cylinder that exactly contains it) important enough to have it carved on his tomb.
Reach for lateral area whenever the top and bottom are not really "surface" in the sense you care about — a label wrapping a can, paint on the outer wall of a cylindrical tank with no roof, or the fabric of a lampshade. Reach for total area whenever every face is real material — wrapping paper for a gift, sheet metal to fabricate a fully enclosed box, or the paint needed to coat a solid sphere completely. The cube and box shapes on this page only report total area for exactly this reason: a fully closed box has no meaningful "lateral-only" version, since every one of its six faces is equally a side.
There is no shortcut that skips a dimension: knowing only a shape's volume is not enough to recover its surface area, because the same volume can be packaged into many different shapes with very different surface areas — a long thin box and a near-cube can enclose an identical volume while having very different amounts of surface. You need to first solve for the shape's defining measurement (a cube's edge from its volume is the cube root; a sphere's radius from V = ⁴⁄₃πr³ is ³√(3V÷4π)) using the volume calculator, then feed that measurement into this page.
Total surface area counts every face. Lateral counts only the sides, leaving out the top and bottom. The distinction matters more often than it is taught: labelling a tin needs the lateral area, painting a room’s walls needs lateral, and a cylindrical tank standing on the ground needs lateral plus one end rather than both.
| Solid | Total surface area |
|---|---|
| Cube | 6a² |
| Rectangular box | 2(lw + wh + lh) |
| Cylinder | 2πr(r + h) |
| Sphere | 4πr² |
| Cone | πr(r + l), where l = √(r² + h²) |
A cone’s curved surface depends on the slant height l — the distance from the rim to the apex along the surface — not the vertical height h. They are related by Pythagoras: l = √(r² + h²). A cone of radius 3 and height 4 has a slant height of exactly 5, giving a total surface area of 75.3982.
The surface area of a sphere, 4πr², is exactly four times the area of the circle you get by slicing through its middle. Archimedes proved it, considered it his finest result, and asked for the sphere-and-cylinder diagram on his tomb.
Whenever the ends are not part of the job. Labelling a tin, painting the walls of a room, or calculating the material for a pipe all use lateral area. A closed tank being painted inside and out uses total.
Because the curved surface is measured along the slope, not vertically. Unrolled, it forms a sector of a circle whose radius is the slant height. Using the vertical height instead understates the area.
A sphere has exactly two thirds the volume and two thirds the total surface area of the smallest cylinder that contains it. This is Archimedes’ theorem and it holds for any radius.
For a sphere or cube you can work backwards, since one dimension determines everything: find the radius or edge from the volume, then apply the surface formula. For a cylinder or cone you cannot — the same volume can be achieved by many different combinations of radius and height, each with a different surface area.
For the capacity rather than the skin, use the Volume Calculator. Flat shapes are handled by the Area Calculator and the Circle Calculator, and the slant-height step is just the Pythagorean theorem.