Surface area calculator
Pick a shape and enter its sizes, get the surface area.
How do I find the surface area of a cylinder?
2πr² for the two ends plus 2πrh for the side: 2πr(r + h). A can 3 in across the radius and 5 in tall has 2π × 3 × 8 = 150.8 in².
What is the surface area of a sphere?
4πr². A ball with a 2 in radius has 4π × 4 = 50.3 in² of surface.
How do I find the surface area of a box?
2(lw + lh + wh) — the six faces in three pairs. A 2 × 3 × 4 box has 2 × (6 + 8 + 12) = 52 square units.
What about a cone?
πr² for the base plus πrs for the side, where s = √(r² + h²) is the slant height. For r = 3 and h = 4, s = 5 and the total is 24π = 75.4.
What is lateral surface area?
The sides only, without the top and bottom — useful for labels on a can or siding on a round tower.
What is the difference between surface area and volume?
Surface area is the amount of outer skin, measured in square units; volume is the space inside, in cubic units. Wrapping paper, paint and siding depend on surface area, while filling a tank depends on volume. Shapes with the same volume can have very different surface areas: a cube with a volume of 1 has 6.00 units of surface, while a sphere of the same volume has only 4.84, the least possible for that volume.
Do I use the diameter or the radius?
The radius, which is half the diameter. Entering the diameter by mistake makes the answer four times too large for a sphere: a ball with a 6 inch diameter has a radius of 3 inches and a surface of 113.1 square inches, not 452.4.
The formulas and what they include
Each formula adds up the area of every face or curved surface. A box has three pairs of rectangles. A cylinder is two circles plus a rectangle that wraps around, 2πrh, because the side unrolls to a rectangle as wide as the circumference. A cone is a circle plus a curved side of area πrs, where s is the slant height, the distance along the surface from the rim to the tip. A square pyramid is the base plus four triangles of slant height s. A hemisphere with its flat base included is 3πr²: a curved part 2πr² plus a circle πr². "Lateral" means the sides only, with no top or bottom.
Worked example: painting a tank and wrapping a box
A cylindrical tank with a radius of 4 ft and a height of 10 ft has a side of 2π × 4 × 10 = 251.3 square feet and a total of 351.9 square feet with both ends. If only the outside wall is painted, the lateral figure is the one you need. A box 12 by 10 by 8 inches has 2 × (12 × 10 + 12 × 8 + 10 × 8) = 592 square inches of surface, and 352 of that is the four sides. For a cone with a 5 inch radius and a 12 inch height the slant height is √(5² + 12²) = 13 inches, the side is 204.2 square inches, and the total with the base is 282.7.
Common mistakes
Using the vertical height instead of the slant height for a cone or pyramid understates the side area. Mixing units is a second problem: a length in feet and another in inches must be converted before you multiply, and the answer is in square units of whatever you use. Confusing total and lateral area leads to ordering too little or too much material. A third is forgetting that real objects are often open at one end, such as a cup or a tank without a lid, and that seams, overlaps and waste add to the amount of material needed beyond the geometric surface area.
Sources: surface-area formulas for the prism, cylinder, cone, sphere and pyramid as in OpenStax Prealgebra 2e (geometry, section 9.6 on volume and surface area) and OpenStax Elementary Algebra 2e; the sphere has the least surface area for a given volume (the isoperimetric inequality in three dimensions). Related: volume calculator, area calculator and circle calculator.