Surface Area Calculator
Pick a shape and type its sizes in any mix of units to get the total, side and base area, plus the volume and the working. Advanced adds paint or wrap needed and finds a missing size.
The size you pick is worked out from this area (and the other size), and filled in below.
Working: the sides
Working: total surface area
How your surface area is worked out
Put every size in one unit, find the sides, add the ends, then use the area for volume or paint.
- 1
Sizes in cm
r = 5, h = 20 cm
- 2
The curved side
2 × π × r × h2 × π × 5 × 20= 628.319 cm²
- 3
Add the ends
A = 2πrh + 2πr²628.319 + 2 × 78.5398= 785.398 cm²
- 4
Bonus: the volume
Volume= 1,570.8 cm³= 1.5708 litres
Read more: every formula, a worked example and paint tips
Surface area is the total area of the outside of a solid, measured in square units such as cm², m² or ft². It's what you would paint, wrap or cover. The side (lateral) area leaves out the base and top; the base area is one flat end. Every size must be in the same unit first, so the calculator changes them all to the unit you used most.
The formula
cube = 6 × side² · sides = 4 × side²
box = 2 × (length × width + length × height + width × height) · sides = 2 × height × (length + width)
sphere = 4 × π × radius²
hemisphere = 2 × π × r² for the dome + π × r² for the flat base = 3 × π × r²
cylinder = 2 × π × r × height for the side + π × r² for each end (2 ends closed, 1 for an open top, 0 for a tube)
cone = π × r × slant + π × r², slant = √(r² + height²)
square pyramid = side² + 2 × side × slant, slant = √((side ÷ 2)² + height²)
triangular prism = 2 × triangle + (a + b + c) × length, triangle = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2
capsule = 2 × π × r × middle length + 4 × π × r²
torus = 4 × π² × ring radius × tube radius
frustum = π × (R + r) × slant + π × R² + π × r², slant = √((R − r)² + height²)
ellipsoid ≈ 4 × π × (((ab)ᵖ + (ac)ᵖ + (bc)ᵖ) ÷ 3)^(1/p), p = 1.6075
paint = area × coats × (1 + waste %) ÷ coverage
Worked example
A closed cylinder with a 5 cm radius and 20 cm height. The curved side is 2 × π × 5 × 20 = 200π = 628.319 cm². Each end is π × 5² = 25π = 78.5398 cm², so the base area is 78.5398 cm². The total is 628.319 + 2 × 78.5398 = 250π = 785.398 cm². With an open top it would be 628.319 + 78.5398 = 706.858 cm², and as a tube with no ends just 628.319 cm². Its volume is π × 5² × 20 = 500π = 1,570.80 cm³, or 1.5708 litres.
To paint the whole outside with 2 coats and 10% extra: 785.398 cm² is 0.0785398 m², and 0.0785398 × 2 × 1.10 = 0.172788 m² of painting. At 10 m² per litre that's 0.172788 ÷ 10 = 0.0172788 litres, about 17.3 mL.
Area units
| Unit | In square metres | Note |
|---|---|---|
| 1 mm² | 0.000001 | 1/100 of a cm² |
| 1 cm² | 0.0001 | 100 mm² |
| 1 in² | 0.00064516 | 6.4516 cm² |
| 1 ft² | 0.09290304 | 144 in² |
| 1 yd² | 0.83612736 | 9 ft² |
| 1 m² | 1 | 10,000 cm², about 10.7639 ft² |
These are exact definitions (the inch is exactly 2.54 cm).
Paint coverage
Typical interior wall paint covers about 10 m² per litre per coat, often printed as 8–12 m²/L or 350–400 ft² per US gallon (10 m²/L is about 407 ft² per US gallon). Primers, textured or porous surfaces (bare wood, brick, new plaster) and dark-to-light colour changes need more paint or an extra coat. These are rule-of-thumb figures, not one maker's spec; always use the coverage printed on your tin. Checked October 2026.
Notes
- The ellipsoid's surface area has no simple exact formula. Knud Thomsen's approximation is within about 1.06% of the exact value, so it's marked approximate (≈). With all three semi-axes equal it is a sphere and exact.
- A sphere, torus and ellipsoid have no flat base or straight sides, so the whole surface is curved and the side area equals the total.
- A cylinder "tube" counts only the outside of the curved side. For a pipe with a wall, use the volume calculator's tube shape, which also counts the inside surface and the ring-shaped ends.
- "A missing size" works backwards from the area: for a cube side = √(A ÷ 6); for a sphere r = √(A ÷ 4π); for a closed cylinder h = (A − 2πr²) ÷ 2πr, or r = (−h + √(h² + 2A ÷ π)) ÷ 2.
Common questions
How do I work out surface area?
Find the area of every face or curved part on the outside and add them up, with all sizes in the same unit. A box 30 × 20 × 10 cm has two faces of each size: 2 × (600 + 300 + 200) = 2,200 cm². Curved shapes have their own formula, such as 4 × π × r² for a sphere.
What is the surface area of a cylinder?
A closed cylinder is 2 × π × r × h for the curved side plus 2 × π × r² for the two ends. With a 5 cm radius and 20 cm height that's 628.319 + 157.080 = 785.398 cm². An open-top cup counts only one end (706.858 cm²) and a tube with no ends only the side (628.319 cm²).
What is the difference between lateral and total surface area?
Lateral area is the sides only: the curved side of a cylinder or cone, or the four walls of a box. Total surface area adds the base and top. For walls you would paint in a room, or a label round a can, the lateral area is the one you want.
How do I find the surface area of a cone?
Work out the slant height first: s = √(r² + h²). The curved side is π × r × s and the base is π × r². A cone with a 5 cm radius and 12 cm height has a slant of 13 cm, a curved side of 65π = 204.204 cm² and a total of 90π = 282.743 cm².
How much paint do I need for a surface?
Multiply the area by the number of coats, add 10% or so for waste, then divide by the coverage on the tin. A typical wall paint covers about 10 m² per litre (roughly 350–400 ft² per US gallon), but check your tin: rough or porous surfaces take more.
Why is the ellipsoid answer marked approximate?
There's no simple exact formula for the outside of an egg-shaped ellipsoid. The calculator uses Knud Thomsen's formula, which is within about 1.06% of the true value. When all three semi-axes are equal it's a sphere and the answer is exact.
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