球体与圆柱体计算器

The Sphere/Cylinder/Cone Calculator handles the three most common rounded 3D shapes — they all share the same circular cross-section profile, which is why one calculator can solve all three. Pick the shape, give the radius (and the height if you chose cylinder or cone), and the calculator returns the volume, surface area, and other derived quantities. This tutorial walks through what each formula means, why the cone's volume is exactly one-third of the cylinder's, and how to spot the common mistakes.

The three shapes, side by side

The cone's surface area uses slant height ℓ, not vertical height. The slant height is the distance from the apex to a point on the rim along the surface — it relates to vertical height by another Pythagorean relationship: ℓ = √(r² + h²).

The sphere

A sphere is fully determined by its radius. There is no "height" — a sphere is symmetric in every direction.

Volume: V = (4/3)πr³. The volume scales as the cube of the radius. A sphere with twice the radius has 8× the volume.

Surface area: SA = 4πr². The surface area scales as the square of the radius. A sphere with twice the radius has 4× the surface area.

Interesting fact: a sphere has the smallest surface area of any shape enclosing a given volume. This is why soap bubbles form spheres (minimum surface tension energy) and why raindrops are roughly spherical in free fall.

Example: a basketball has radius ≈ 12 cm. Volume = (4/3)π(12)³ = (4/3)π(1728) ≈ 7238 cm³. Surface area = 4π(12)² = 576π ≈ 1810 cm².

The cylinder

A cylinder is two parallel circular bases joined by a curved lateral surface. It needs two inputs: radius r and height h.

Volume: V = πr²h. This is simply (base area) × (height). Pour water into a cylindrical glass to fill it: the volume of water equals the floor area (πr²) times how high it stands (h). Cylindrical tanks, cans, pipes, and pillars all use this formula.

Surface area has three pieces:

• Top base: πr²
• Bottom base: πr² (same as top)
• Lateral (side) surface: 2πr × h. Unrolling the side flat gives a rectangle with width 2πr (the circumference) and height h.

Total: SA = 2πr² + 2πrh = 2πr(r + h).

For an open cylinder (a pipe or a cup with no lid), use lateral area only or subtract the missing base. Our calculator returns the closed-cylinder total.

Example: a soup can with r = 4 cm, h = 12 cm. V = π(4)²(12) = 192π ≈ 603 cm³. SA = 2π(4)(4 + 12) = 128π ≈ 402 cm².

The cone

A cone has a circular base, an apex (point), and a curved lateral surface joining them. The two inputs are the base radius r and the perpendicular height h (apex straight down to the base).

Volume: V = (1/3)πr²h. Exactly one-third of a cylinder with the same base and same height.

Why exactly one-third? A classic demonstration: a cone and a cylinder with identical bases and identical heights are made out of paper. Filling the cone with sand three times exactly fills the cylinder. The factor of 1/3 isn't arbitrary — it comes out of calculus (integrating the radius² over the height), but the intuitive way to remember it is "the cone tapers, so on average it has half the radius… and (1/2)² × some integration factor works out to 1/3".

Surface area: cone surface area uses the slant height ℓ, which is the diagonal from apex to rim:

• Base: πr²
• Lateral: πrℓ where ℓ = √(r² + h²)

Total: SA = πr² + πrℓ = πr(r + ℓ).

If the problem gives you slant height directly, use it as ℓ. If it gives you perpendicular height, compute ℓ first.

Example: an ice cream cone with r = 2.5 cm, h = 10 cm. Slant height ℓ = √(2.5² + 10²) = √(6.25 + 100) = √106.25 ≈ 10.31 cm. V = (1/3)π(2.5)²(10) = (1/3)(62.5)π ≈ 65.4 cm³. Lateral SA = π(2.5)(10.31) ≈ 80.9 cm².

Open vs closed

Real-world objects often miss a face: an open can has no top, an ice cream cone has no base (otherwise you couldn't eat it). Adjust the surface area by subtracting whichever face is absent:

• Open cylinder (no top, has bottom): SA = πr² + 2πrh
• Open cone (no base): SA = πrℓ
• Hollow tube (cylinder, no top, no bottom): SA = 2πrh

Our calculator returns the closed-shape total — subtract the missing piece by hand if needed.

Hemispheres and segments

A hemisphere is half a sphere. V = (2/3)πr³ and the curved surface is 2πr² (plus πr² for the flat circular base if you need closed). Useful for domes, half-tanks, and bowls.

For partial spheres (spherical caps, spherical zones) the formulas get more involved — those are not in this calculator. See an advanced reference for V = (πh²/3)(3r − h) and similar.

Common mistakes

• Using slant height where you want perpendicular height (or vice versa) for cone volume. Volume needs the perpendicular height h. Surface area needs the slant height ℓ. They are different numbers; only the calculation that demands one accepts that one.
• Forgetting the 1/3 in cone volume. Without it your answer is 3× too large — exactly the cylinder volume.
• Using diameter as radius. Halve it. If a ball is 24 cm "across", r = 12.
• Confusing surface area and lateral area for a cylinder. Lateral = side only (2πrh). Surface = lateral + 2 bases. A label says "lateral area" or "total surface area" — check which one the problem wants.
• Unit cubes vs squares. Volume is cm³, surface area is cm². Mixing units is a giveaway that something is off.

Where to go next

Related calculators on this site:

• Cube and Rectangular Prism Calculator — for box-shaped solids.
• Cone Formula — the formula-focused page for cone work.
• Cylinder Formula — same, for cylinder.
• All 3D Geometry Formulas — comparison sheet for prism, pyramid, sphere, cylinder, cone, frustum.