Ultimate 3D Shapes Surface Area and Volume Formulas Chart
Geometry can feel like a maze, but every 3D shape is just a puzzle of length, width, and height. This guide is built to be your single reference for all 3D geometry needs—whether you’re tackling Class 9 surface area problems or Class 10 volume proofs.
| 3D Shape | Visual | Formula Variables | LSA / CSA | TSA | Volume |
|---|---|---|---|---|---|
| Cube |  | a = side | 4a2 | 6a2 | a3 |
| Cuboid | | l, b, h | 2h(l+b) | 2(lb+bh+lh) | lbh |
| Cylinder | | r, h | 2πrh | 2πr(r+h) | πr2h |
| Cone | | r, h, l | πrl | πr(l+r) | 13 πr2h |
| Sphere | | r | 4πr2 | 4πr2 | 43 πr3 |
| Hemisphere | | r | 2πr2 | 3πr2 | 23 πr3 |
| Frustum (Cone) | | R, r, h, l | πl(R+r) | πl(R+r) + π(R2+r2) | 13 πh(R2+r2+Rr) |
| Hollow Cylinder | | R, r, h | 2πh(R+r) | 2π(R+r)(h+R-r) | πh(R2-r2) |
| Hollow Sphere | | R, r | 4π(R2+r2) | 4π(R2+r2) | 43 π(R3-r3) |
| Triangular Prism | | b, h, L | (a+b+c)L | (a+b+c)L + 2(Area) | Area of base × L |
| Tetrahedron | | a = edge | 3a2 | √3a2 | a36√2 |
Deep Dive into 3D Shapes
The Frustum: The Cut-Off Cone
Imagine you have an ice cream cone and you slice off the pointy bottom part. What remains is a Frustum. It is common in Class 10 geometry. The volume formula looks intimidating—1/3 πh(R²+r²+Rr)—but it is essentially the volume of the original large cone minus the volume of the small cone you cut off.
Hollow Cylinder: The Pipe Logic
Think of a water pipe. It has an inner radius (r) and an outer radius (R). The volume of the material is just the volume of the big cylinder minus the empty hole in the middle. The formula πh(R²-r²) perfectly represents this difference.
Tetrahedron: The Simplest Pyramid
A tetrahedron is a pyramid with a triangular base and three triangular faces. It is the simplest 3D shape that is not “flat.” Every edge a is equal. It is often used in architecture and molecular modeling for its extreme structural stability.
Real-World Logic: Why Formulas Matter
Surface Area isn’t just a number; it’s the amount of paint needed for a wall, the amount of sheet metal for a tank, or the amount of fabric for a tent. Volume is the capacity—how much liquid a tank holds, how much sand fills a box, or how much space a room has.
Tips for Mastering These Formulas
- Practice Units: Always convert all dimensions (length, radius, height) to the same unit before calculating.
- Slant Height: When in doubt for cones, use Pythagoras: l = √(r²+h²).
- Draw it Out: Even a rough sketch of the cross-section helps you identify if the shape is hollow or solid.
- Visualize “Unfolding”: Surface area is really just the area of the flattened-out shape.