Surface Area Calculator
Calculate the total, lateral and base surface area of a cube, prism, cylinder, cone, sphere, hemisphere, pyramid, capsule or cone frustum. Pick a shape, enter its dimensions and get the area in the unit you choose.
Find the Surface Area of Common 3D Shapes
Surface Area Formula
Where:
- \(A_t\) — total surface area (all faces of the solid)
- \(a\), \(b\), \(c\) — edges of the cube, rectangular prism, pyramid base or equilateral triangle side
- \(r\) — radius of the cylinder, cone, sphere, hemisphere or capsule (top radius of the frustum)
- \(R\) — bottom (larger) radius of the cone frustum
- \(h\) — height of the solid (the cylinder length of the capsule)
- \(l\) — slant height along the sloped surface
The total area is split into the lateral (curved or side) surface and the flat base(s). For every shape except the sphere and the capsule, the calculator reports \(A_t\), \(A_l\) and \(A_b\) separately, so you can use only the part you need, for example when painting walls but not the floor.
Surface Area – Calculation Example
Example: a closed cylindrical tank with radius \(r = 5\) cm and height \(h = 12\) cm — the calculator's default values.
\[ A_l = 2\pi r h = 2\pi \times 5 \times 12 = 120\pi \approx 376.991 \text{ cm}^2 \]
\[ A_b = 2\pi r^2 = 2\pi \times 5^2 = 50\pi \approx 157.08 \text{ cm}^2 \]
\[ A_t = A_l + A_b \approx 376.991 + 157.08 = 534.071 \text{ cm}^2 \]
Result: the total surface area is about 534.071 cm² (0.0534 m², roughly 82.76 in²). The curved side of the cylinder alone is 376.991 cm² and the two circular ends add 157.08 cm².
How to use the calculator
- Pick the solid in the Shape dropdown — cube, rectangular prism, cylinder, cone, sphere, hemisphere, square pyramid, triangular prism, capsule or cone frustum. The form automatically shows only the dimensions the chosen shape needs.
- Choose the Length unit (mm, cm, m, in, ft or yd). Enter all dimensions in that unit; areas are returned in the same unit squared.
- Fill in the visible Dimensions fields: Edge / side (a), Side (b) and Side (c) for box-like shapes, Radius (r) and Height (h) for round shapes, and Bottom radius (R) for the frustum.
- For a hemisphere, decide whether to tick Include flat base — this adds the circular disk \(\pi r^2\) to the total.
- Read the results: Total surface area, Lateral / curved area, Base area(s), plus handy conversions to cm², m², in² and ft², the formula with your values substituted, and a labelled diagram of the shape.
What is surface area?
Surface area is the total area of all the faces or curved surfaces that enclose a three-dimensional object — the amount of material you would need to wrap the shape completely. It is measured in square units: if the dimensions are in centimetres, the area comes out in square centimetres (cm²).
The calculator separates the area into three parts, and the distinction matters in practice. The lateral (or curved) area covers just the sides or the curved wall of the solid — think of the label wrapped around a can. The base area(s) are the flat ends: the two circles of a cylinder, the single square under a pyramid. Adding them gives the total surface area. When you estimate paint for the outside of a water tank, you need the total; when you estimate wallpaper for a living room, only the lateral area of its walls counts.
Surface area is closely related to volume, but they behave differently. Doubling every dimension of a shape multiplies its volume by 8, yet its surface area only by 4 — because area grows with the square of the scale factor, not the cube. This is why small objects lose heat quickly (high area relative to volume), and why grinding coffee into fine powder makes it brew faster.
The conversions to cm², m², in² and ft² are useful when shopping for materials: fabric is often sold by the square metre, paint coverage is rated in square feet per litre, and packaging labels may use any of the four.
Assumptions, limitations and tips
- The calculator handles ideal, regular shapes. Real objects have seams, rims and rounded edges, so add 5–10% to the result when ordering material such as fabric, foil or paint.
- The triangular prism assumes an equilateral triangle base with side \(a\); a prism with a different triangle needs the general formula \(A_t = (\text{perimeter}) \times h + 2 \times (\text{triangle area})\). A triangle area calculator can supply the base area.
- The square pyramid uses the true slant height \(l\) of the face, not the edge from apex to corner, and requires a square base; the capsule height \(h\) is only the straight cylinder part, not the tip-to-tip length.
- For the hemisphere, remember to set Include flat base according to your situation — a dome standing on the floor needs the base disk counted only if the floor itself is part of the surface.
- Always enter positive numbers, and in the cone frustum keep the bottom radius \(R\) greater than or equal to the top radius \(r\).
- For cylinders, spheres and cones you can explore further with the dedicated cylinder surface area, sphere volume and surface area and cone volume and surface area calculators.
Frequently Asked Questions
How do I find the surface area by hand?
Start by writing down the formula for your shape, substitute the measurements, and do the arithmetic step by step. For a cylinder, for example, compute \(A_l = 2\pi rh\) for the side and \(A_b = 2\pi r^2\) for the ends, then add them. Use 3.14159 for \(\pi\) or the \(\pi\) button on a calculator, and keep all lengths in the same unit before squaring.
Why is my answer wrong when I calculate a cone or pyramid?
The most common slips are using the diameter instead of the radius and confusing height with slant height. The curved surface of a cone needs the slant height \(l = \sqrt{r^2 + h^2}\), not the vertical height \(h\) — the slant is always longer. For a square pyramid, the same applies: use the distance from the apex to the middle of a base edge, measured along the face.
How do I convert between square units like cm², m² and ft²?
Square the length conversion factor. Since 1 m = 100 cm, 1 m² = 10 000 cm²; since 1 ft = 12 in, 1 ft² = 144 in². To go from cm² to in², divide by 6.4516, and to go from m² to ft², multiply by about 10.764. This calculator performs all of these conversions automatically after choosing your input unit.
Should I use total or lateral surface area for painting or wrapping?
Use the total area only when every face is actually covered; otherwise the lateral area is what you need. When painting a room or the outside wall of a tank, the floor and ceiling usually don't count, so lateral is right. Wrapping a closed gift box, insulating a full pipe including its ends, or coating a ball requires the total.
When does the hemisphere's flat base count?
Count the \(\pi r^2\) disk whenever the flat circular face is part of the surface you're covering — for example painting a bowl inside and out. Skip it for a dome sitting on the floor, where the base is the ground itself. The curved dome alone is always \(2\pi r^2\), and ticking "Include flat base" in the calculator adds the disk.
My object isn't a perfect shape. Can I still use this calculator?
Yes, approximate it by breaking it into pieces. A hot dog, pill or rounded container works as a capsule; a lampshade or truncated post as a cone frustum; a gazebo roof as a cone or pyramid. Calculate each part separately and add the areas, ignoring small overlaps or seams. For ordering material, add a 5–10% margin for joins, folds and cutting waste.
How is surface area different from volume?
Surface area measures the outside shell in square units; volume measures the space inside in cubic units. They scale differently: doubling every dimension quadruples the area \((2^2)\) but multiplies the volume by eight \((2^3)\). If you need capacity rather than coverage — how much a tank holds rather than how much paint its walls need — use the volume calculator instead.