Find the volume and surface area of a standard basketball (radius ~12 cm).
Every 3D solid has a volume (how much space it fills or holds) and a surface area (how much material it would take to cover its outside). This tool covers the six shapes taught in most school geometry courses — cube, cuboid, sphere, hemisphere, cylinder, and cone — and draws a labeled diagram of whichever one you pick.
For the curved shapes (hemisphere, cylinder, cone), the tool shows both the curved surface area (just the curved part) and the total surface area (curved part plus the flat circular base or bases) — a distinction that trips a lot of people up, since which one a problem is actually asking for changes the answer completely.
The formulas: Cube — V = s³, SA = 6s². Cuboid — V = lwh, SA = 2(lw + wh + hl). Sphere — V = (4/3)πr³, SA = 4πr². Hemisphere — V = (2/3)πr³, curved SA = 2πr², total SA = 3πr². Cylinder — V = πr²h, curved SA = 2πrh, total SA = 2πr(r + h). Cone — V = (1/3)πr²h, curved SA = πrl, total SA = πr(r + l), where l is the slant height.
Worked example (cone, r=3, h=4): first find the slant height using the Pythagorean theorem, l = √(r² + h²) = √(9 + 16) = √25 = 5. Then volume = (1/3) × π × 3² × 4 ≈ 37.70, curved surface area = π × 3 × 5 ≈ 47.12, and total surface area = π × 3 × (3 + 5) ≈ 75.40. A common mistake is using the cone's height (h) in place of its slant height (l) when computing surface area — they're only the same length if the cone is flattened, which a real cone never is.
Curved surface area (CSA) is just the curved part — the outside of a cylinder’s tube, not its lids. Total surface area (TSA) adds the flat circular base(s) on top. A cube and cuboid don’t have this distinction since all their faces are flat, so only one surface area is shown for those two.
Volume is simply half of a full sphere’s volume. But surface area isn’t just half a sphere’s surface area — slicing a sphere in half removes the curved surface on that cut face and replaces it with a new flat circular face, so the curved surface area is exactly half a sphere’s (2πr²), while the total surface area adds that new flat circle back in (3πr² total).
It’s the straight-line distance from the tip of the cone to the edge of its base circle — not the same as the cone’s vertical height. By the Pythagorean theorem, slant height = √(radius² + height²), since the radius, the height, and the slant form a right triangle.