Surface Area Calculator
Calculate the surface area of 8 three-dimensional shapes. The same calculator also provides volume and additional measurements like slant height and diagonal.
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How It Works
Each face is a square with area s². Six faces total. SA = 6s². The space diagonal is s√3.
SA = 6s²s=4 → SA=96Three pairs of rectangles: top+bottom (lw), front+back (lh), left+right (wh). SA = 2(lw+lh+wh).
SA = 2(lw + lh + wh)6×4×3 → SA=108SA = 4πr². The sphere has the minimum surface area for a given volume — hence bubbles and drops are spherical.
SA = 4πr²r=5 → SA=314.16Two circular caps: 2πr². Lateral rectangle: 2πrh (circumference × height). Total: SA = 2πr(r+h).
SA = 2πr(r+h)r=3, h=8 → SA=207.3Base circle: πr². Lateral surface: πrl where l=slant height=√(r²+h²). Total: SA = πr(r+l).
SA = πr(r + l), l=√(r²+h²)r=4, h=3 → l=5, SA=113.1Square base: b². Four triangular faces each with area = b×l/2 where l = slant height. Total: SA = b² + 2bl.
SA = b² + 2bl, l=√((b/2)²+h²)b=6, h=4 → SA=84Quick Reference
Common examples — verify instantly above.
Cube
s=4
SA=96
Cube
s=5
SA=150
Sphere
r=5
SA=314.2
Cylinder
r=3, h=8
SA=207.3
Cone
r=4, h=3
SA=113.1
Pyramid
b=6, h=4
SA=84
Box
6×4×3
SA=108
Box
5×5×5
SA=150
Tips & Shortcuts
Surface area of a cylinder: think of it as unrolling — two circles (2πr²) and a rectangle (2πrh).
A sphere has the smallest surface area for a given volume. This is why bubbles are round.
Slant height of a cone: l = √(r²+h²). Use this for lateral surface area, not for volume.
For a cube vs rectangular prism with equal volume: the cube always has the smaller surface area.
Lateral surface area excludes bases. Total surface area includes all faces. Know which is required.
Surface area scales with the square of dimensions. Doubling all lengths quadruples surface area.
Common Mistakes
Computing lateral surface area instead of total
Lateral area = sides only. Total surface area includes top and bottom bases. Always check which is required.
Using vertical height instead of slant height for cone lateral area
Cone lateral area = πrl where l = slant height = √(r²+h²), not the vertical height h.
Forgetting to multiply by 2 for top and bottom of cylinder
Cylinder SA = 2πr² (both caps) + 2πrh (lateral). Many forget the factor of 2 for the caps.
Computing volume when surface area is asked
Volume is in cubic units; surface area is in square units. Read the question carefully before calculating.
Using diameter in the sphere formula instead of radius
SA = 4πr². If given diameter d, use r = d/2. Using diameter directly gives 4× the correct answer.
Forgetting all four triangular faces of a pyramid
A square pyramid has 4 triangular sides plus 1 square base. All 5 faces contribute to total surface area.
Frequently Asked Questions
Surface area is the total area of all outer faces of a 3D shape, measured in square units.
SA = 4πr². A sphere has the smallest surface area for a given volume.
Lateral (side) surface area excludes the top and bottom bases. Total surface area includes all faces.
SA = 2πr² + 2πrh = 2πr(r+h). Two circular caps (2πr²) plus the lateral rectangle (2πrh).
Surface area measures outer faces in square units. Volume measures enclosed space in cubic units.
Slant height l = √(r²+h²) — the distance along the side from base edge to apex. Used in lateral surface area = πrl.
Total surface area includes every face. Lateral surface area includes only the sides, excluding the base(s). For a cylinder: lateral SA = 2πrh (curved side only), total SA = 2πrh + 2πr² (adding both circular bases). For a cone: lateral SA = πrl (where l is slant height), total SA = πrl + πr² (adding the base). Lateral SA is relevant for painting tank walls when the caps are manufactured separately.
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