Maths⏱ 5 min read
Surface Area of 3D Shapes: Every Formula You Need
Surface area calculations are needed for painting, wrapping, insulating, and material estimation. Here are the formulas for every common shape with worked examples.
Surface area is the total area of all faces of a 3D object. Unlike volume (which fills the space inside), surface area tells you how much material you need to cover the outside — paint, insulation, packaging, or fabric.
Cube and Rectangular Box (Cuboid)
Cube (all sides equal length a):
Surface area = 6a^2
Cube with side 4m: SA = 6 x 16 = 96 m2
Cuboid (length L, width W, height H):
Surface area = 2(LW + LH + WH)
Box: 5m x 3m x 2m
SA = 2(5x3 + 5x2 + 3x2) = 2(15 + 10 + 6) = 2(31) = 62 m2
Practical: painting a room (4 walls + ceiling):
= 2(LH + WH) + LW (ceiling only)
= 2(5x2.4 + 4x2.4) + 5x4
= 2(12 + 9.6) + 20
= 43.2 + 20 = 63.2 m2
Sphere and Hemisphere
Sphere (radius r):
Surface area = 4πr^2
Sphere radius 3m: SA = 4 x 3.14159 x 9 = 113.1 m2
Hemisphere (half sphere):
Curved surface = 2πr^2
Total (including flat base) = 3πr^2
Practical: coating a dome with radius 5m:
Curved surface = 2 x 3.14159 x 25 = 157.1 m2
Cylinder
Cylinder (radius r, height h):
Curved surface = 2πrh
Two circular ends = 2πr^2
Total surface area = 2πr(r + h)
Water tank: radius 0.5m, height 1.2m
Curved: 2 x 3.14159 x 0.5 x 1.2 = 3.77 m2
Two ends: 2 x 3.14159 x 0.25 = 1.57 m2
Total: 5.34 m2
Open cylinder (no top lid — like a bucket):
SA = 2πr^2 + 2πrh - πr^2 = πr^2 + 2πrh
Cone
Cone (base radius r, height h, slant height l):
Slant height: l = sqrt(r^2 + h^2)
Curved surface = πrl
Base circle = πr^2
Total = πr(r + l)
Example: cone r = 3m, h = 4m
l = sqrt(9 + 16) = sqrt(25) = 5m
Curved surface = 3.14159 x 3 x 5 = 47.12 m2
Total: 47.12 + 28.27 = 75.40 m2
Triangular Prism (Common in Architecture)
Triangular prism (triangle base with sides a,b,c, length L):
SA = (perimeter of triangle x L) + 2 x triangle base area
For a right-angled triangle (sides 3, 4, 5):
Triangle area = (3 x 4) / 2 = 6
Perimeter = 3 + 4 + 5 = 12
Prism length L = 8m
SA = (12 x 8) + (2 x 6) = 96 + 12 = 108 m2
Practical: pitched roof (triangular cross-section):
SA = total area of roof panels to tile/felt