Geometry: circle, cylinder, cone, sphere as one live transformation
Calculate area, circumference, volume and surface area for a circle, cylinder, cone or sphere - with one drawing that shows how the four shapes connect.
Cylinder volume is V = π · r² · h: the circular base area times the height. A cylinder with radius 5 cm and height 10 cm holds 785.4 cm³, about 0.79 litres. Pick a shape, enter the radius, and you get area, circumference, volume or surface area, depending on whether you are working with a circle, cylinder, cone or sphere.
The 2D base. Pull it up, taper it, or spin it - that's where the three bodies come from.
Radius: 5 cm · Diameter: 10 cm
Show formula
A = π · r2 = π · 52 = 78.54 cm²
U = 2 · π · r = 2 · π · 5 = 31.42 cm
d = 2 · r = 2 · 5 = 10 cm
How are a circle, cylinder, cone and sphere connected?
The four shapes are not four separate topics. They are one family built on the same circle, and the drawing in the calculator makes that visible: one circle, three transformations. Most calculators split these into four separate pages, so the thread connecting them gets lost.
- Circle - the flat 2D base. Just a radius, no height.
- Cylinder - the circle pulled straight up (extruded). Same base, plus a height.
- Cone - the circle pulled up to a single point. Same base, a tip instead of a lid.
- Sphere - the circle revolved around its own axis. No height needed, just the radius.
Because all four share the same circle, the circle area π · r² turns up inside every formula. For the angular relatives - triangle, prism, pyramid - the same idea lives in the sister tool Geometry: angular shapes.
What are the formulas behind each shape?
Every formula builds on the circle area A = π · r² and the circumference U = 2 · π · r. Click "Show formula" in the calculator and each one appears with your radius and result - most lines also show the substituted middle step, handy when you need to follow the working for homework or a test, not just the final number.
| Shape | Volume | Surface area |
|---|---|---|
| Cylinder | π · r² · h | 2 · π · r² + 2 · π · r · h |
| Cone | ⅓ · π · r² · h | π · r² + π · r · s |
| Sphere | (4/3) · π · r³ | 4 · π · r² |
For the cone, s is the slant height: the distance from the tip down to the edge of the base. It comes straight from the Pythagorean theorem, s = √(r² + h²), and you see it listed separately so you can check the lateral area yourself.
Radius or diameter - which do I enter?
The radius runs from the centre to the edge; the diameter goes all the way across and is exactly twice as long. A 30 cm pizza has a 15 cm radius. Mixing up the two is the single most common mistake in circle maths, because every formula here expects the radius.
So the calculator gives you a toggle: enter either the radius or the diameter, and the other value sits live next to it. That way you never accidentally calculate with double the value. Useful when your problem is stated in diameters but the formulas want the radius.
Surface area, lateral area or base area - what is what?
A cylinder and a cone each have three different areas, and they are easy to confuse. Short version: the surface area is everything together, the lateral area is just the curved side, and the base area is just the bottom disc.
- Base area - the circle at the bottom, π · r². On a cylinder the same circle counts at the top and bottom.
- Lateral area - the curved side wall. On a cylinder it is a rectangle unrolled; on a cone it is a pie slice.
- Surface area - every outer face added together. To paint a whole can you need the surface area; for just the label wrapped around it, the lateral area.
For a cylinder all three values show at once; the drawing colour-separates the base and lateral surfaces, so you do not have to guess which number belongs to which face.
Where do these shapes show up in everyday life?
Round solids sit in almost every household, and the formulas answer concrete questions: how much fits inside, how much material you need, how heavy it will be.
- A tin can or water tank - a cylinder. The volume tells you how many litres it holds; results show directly in litres, m³ and US gallons.
- An ice cream cone or a pile of loose material - a cone. Volume and slant height in one go.
- A globe, ball or marble - a sphere. Surface area for painting it, volume for the weight at a known density.
If you need other number work afterwards, the percentage calculator handles proportions and the large number names tool names very large results.
Frequently Asked Questions
How do you calculate the volume of a cone?
Cone volume is V = ⅓ · π · r² · h: exactly one third of a cylinder with the same base and height. A cone with radius 5 cm and height 10 cm holds ⅓ · π · 25 · 10 = 261.8 cm³. The one-third factor is why an ice cream cone holds far less than it looks.
Why is sphere volume (4/3) · π · r³ and not π · r³?
The 4/3 factor comes from the calculus derivation. Picture it this way: a sphere fills two-thirds of the smallest cylinder it fits inside. You do not have to derive it yourself - the calculator substitutes the factor correctly. What matters is that for sphere volume the radius is cubed, not squared.
Can I work in different units?
Yes. The unit switcher offers millimetres, centimetres, metres, inches and feet, and the result converts instantly. For volumes you also get a parallel readout in litres, cubic metres and US gallons, so you do not have to convert in your head.