Surface Area Calculator,
7 shapes, any unit.

Geometry guide

Surface area formulas for every common 3-D shape.

The surface area of a three-dimensional solid is the total area of all its outer faces, the amount of material you would need to wrap it completely. It is measured in square units (cm², m², ft², etc.) and differs from volume, which measures the space enclosed inside. Surface area is critical in engineering, manufacturing, biology, and everyday life: from calculating how much paint covers a room to modelling how a cell exchanges gases across its membrane.

Total, lateral, and base surface area

For curved or prismatic shapes it is useful to distinguish three components:

• Total surface area (TSA): the sum of every face. What you need to paint or wrap the entire solid.
• Lateral surface area (LSA): only the side or curved surface, excluding the base(s). Useful when the shape sits on a flat base (e.g., a can on a shelf) or when two identical shapes are stacked (their shared faces need no coating).
• Base surface area: only the top and/or bottom face(s). For a cylinder or prism with two parallel bases this counts both.

TSA = LSA + base area. For a sphere (no flat faces), total and lateral surface area are the same.

Complete formula reference

Sphere: SA = 4πr²

A sphere's surface area is elegantly simple: four times the area of a great circle. Archimedes proved this in the 3rd century BC, showing that the surface of a sphere equals the lateral surface of the circumscribed cylinder. For a basketball with radius 12 cm:

Cylinder: SA = 2πr(r + h)

Think of unrolling a cylinder: the lateral surface becomes a rectangle with width = 2πr (the circumference) and height h. Two circles of area πr² cap the ends:

Choosing the right formula for your application matters: when calculating how much label wraps around a can, use lateral SA. When calculating how much sheet metal to cut, use total SA.

Cone: SA = πr² + πrl

A cone's lateral surface unfolds into a sector of a circle with radius equal to the slant height l. The slant height is the straight-line distance from the apex to the rim of the base, computed from the perpendicular height h and base radius r using the Pythagorean theorem:

For a party hat with radius 6 cm and height 8 cm: l = √(36 + 64) = 10 cm, TSA = π × 6 × (6 + 10) ≈ 301 cm².

Rectangular prism: SA = 2(lw + lh + wh)

A rectangular prism (cuboid) has three pairs of identical rectangular faces. Add each pair and multiply by 2:

For a gift box 30 cm × 20 cm × 10 cm: SA = 2(600 + 300 + 200) = 2200 cm² of wrapping paper needed.

Square pyramid: SA = b² + 2bl

A square pyramid has a square base and four congruent triangular faces. The area of each triangle is ½ × base × slant height (not perpendicular height). The slant height runs from the midpoint of a base edge to the apex:

Triangular prism: equilateral base

This calculator uses an equilateral triangular base (all three sides equal to a) for simplicity. The two triangular faces and three rectangular side faces contribute:

Real-world applications of surface area

• Architecture & construction: Calculating how much paint, plaster, or cladding is needed for walls, ceilings, and facades. A 4 m × 3 m × 3 m room has about 66 m² of paintable surface (walls + ceiling), requiring roughly two coats of a 10 m²/L paint, about 13 litres.
• Manufacturing & packaging: Determining how much sheet metal, cardboard, or plastic is needed to form a container. The lateral SA gives the strip of metal to roll a cylinder; the total SA gives the blank area before stamping the ends.
• Heat transfer: Heat exchangers, radiators, and cooling fins are designed to maximise surface area relative to volume. A higher SA/V ratio means faster heat exchange, which is why radiator fins are thin and numerous.
• Biology: Cell size is limited partly by the SA/V ratio — a cell must absorb nutrients and expel waste across its membrane fast enough to sustain its volume. As cells grow, SA grows as r² but volume as r³, so large cells must be flatter (red blood cells, neurons) or highly folded (intestinal villi).
• Astronomy & planetary science: Stefan-Boltzmann law ties radiated power to SA × T⁴. A planet's equilibrium temperature depends on how much surface area faces the star vs. the total area available to radiate heat.

What is slant height and why does it matter?

For cones and pyramids, the slant height is not the same as the perpendicular height. The perpendicular height h goes straight down from the apex to the centre of the base. The slant height l goes from the apex to the edge (cone) or midpoint of an edge (pyramid).

This distinction is critical: using h instead of l in the lateral SA formula will give the wrong answer. When measuring a real cone or pyramid, you may find it easier to measure the slant height directly with a tape rule. If you only have the perpendicular height, the calculator derives the slant height automatically from l = √(r² + h²) (cone) or l = √((b/2)² + h²) (pyramid).

Choosing the right units

Surface area is always expressed in square units. If your dimensions are in centimetres, the result is in cm². The calculator converts automatically — select any of the five supported units (m, cm, mm, ft, in) and all four metric and imperial outputs are computed simultaneously. Common conversions: