Mensuration in 3D focuses on volume (space inside solids) and surface area (total outer area).

Units and interpretation

• Length: cm, m
• Area: cm\(^2\), m\(^2\)
• Volume: cm\(^3\), m\(^3\)

Always keep units consistent before substituting values.

Prisms and cylinders

For any prism/cylinder:

\[ \text{Volume}=\text{base area}\times\text{height}. \]

Cylinder formulas:

\[ V=\pi r^2h,\qquad \text{TSA}=2\pi r^2+2\pi rh. \]

Pyramids and cones

\[ V_{\text{pyramid}}=\frac13(\text{base area})h,\qquad V_{\text{cone}}=\frac13\pi r^2h. \]

Cone curved surface area:

\[ \text{CSA}_{\text{cone}}=\pi rl \]

where \(l\) is slant height. Total surface area adds base area \(\pi r^2\).

Spheres

\[ V=\frac43\pi r^3,\qquad \text{Surface area}=4\pi r^2. \]

Composite solids

Split complex shapes into known solids, then add/subtract volumes or areas appropriately.

• Add volumes when parts are joined.
• Subtract volumes for holes/cut-outs.
• For surface area, remove hidden contact faces.

Volume in context

Common contexts include capacity, material usage, and mass:

\[ \text{mass}=\text{density}\times\text{volume}. \]

Exam strategy

• Sketch and label all dimensions before formula use.
• Choose exact \(\pi\) form first; round at final step if required.
• Convert units early (e.g., cm to m).
• For composite shapes, explain add/subtract logic briefly.