The volume is the space inside a solid; the surface area is the skin around it. A cylinder with a 5 cm radius and a 10 cm height holds π × 5² × 10 = 785.4 cm³ (0.785 L) and has 471.2 cm² of surface. Pick one of seven solids, type its measurements and the calculator gives the volume, the total and side areas, the slant height or diagonal and the capacity in liters and US gallons, with the formulas written out.
The formulas
| Solid | Volume | Surface area |
|---|---|---|
| Cube (edge a) | a³ | 6a² |
| Box (l, w, h) | l × w × h | 2(lw + lh + wh) |
| Cylinder (r, h) | π r² h | 2π r h + 2π r² |
| Cone (r, h) | π r² h ÷ 3 | π r s + π r², s = √(r² + h²) |
| Square pyramid (side a, h) | a² h ÷ 3 | 2a × slant + a² |
| Cone frustum (R, r, h) | π h (R² + Rr + r²) ÷ 3 | π (R + r) s + π (R² + r²) |
| Tube or pipe (R, r, length) | π (R² − r²) × length | the outside, the inside and the two rings |
A cone holds exactly a third of the cylinder with the same base and height, and a pyramid a third of its box. That rule was known to Greek mathematicians more than two thousand years ago and still saves a lot of arithmetic.
One liter, three shapes
How much material does it take to hold one liter? The shape matters:
| Shape holding 1 liter | Size | Surface |
|---|---|---|
| Cube | 10 cm edge | 600 cm² |
| Cylinder as tall as it is wide | 5.42 cm radius, 10.84 cm tall | 553.6 cm² |
| Sphere | 6.20 cm radius | 483.6 cm² |
The sphere needs the least skin for its volume, which is why bubbles and drops are round. Among cylinders, the one whose height equals its diameter uses the least material.
The tube option
For a pipe, the calculator gives two different volumes: the material of the wall, π × (R² − r²) × length, and what the pipe holds inside, π × r² × length. A 20 m garden hose with a 6.5 mm inner radius holds π × 0.65² × 2,000 = 2,655 cm³, about 2.7 liters of water.
Like every zet4u tool, it works offline once the site is installed or saved, so you can check a tank or a planter on site.
Frequently asked questions
How do I find the volume of a cylinder?
Multiply the area of the base by the height: π × r² × h. A can with a 5 cm radius and a 10 cm height holds 785.4 cm³, or 0.785 liters.
How do I find the volume of a cone or a pyramid?
A third of the matching prism: base area × height ÷ 3. A cone has π × r² × h ÷ 3; a square pyramid has side² × h ÷ 3.
What is the difference between surface area and lateral area?
Lateral area is only the sides: the label around a can. Surface area adds the top and the bottom. Paint for the outside of a tank uses the lateral area plus the lid.
How do I find the volume of a bucket or flower pot?
It is a cone frustum: π × h × (R² + R·r + r²) ÷ 3, with R and r the radii at the top and bottom. The calculator also gives the area of an open bucket, without the lid.
How many liters is 1,000 cm³?
Exactly one liter. One cubic meter is 1,000 liters, and one cubic foot is 28.317 liters.
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