Mensuration (3D)
🔒 Log in to trackmedium importance~2 Q in Tier 120 formulas⚡ 10 shortcuts5 subtopics
Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Cube and cuboid
Volume
V=lbh,\qquad V_{\text{cube}}=a^3
Lateral surface
LSA=2(l+b)h,\qquad LSA_{\text{cube}}=4a^2
Total surface
TSA=2(lb+bh+lh),\qquad TSA_{\text{cube}}=6a^2
Space diagonal
d=\sqrt{l^2+b^2+h^2},\qquad d_{\text{cube}}=a\sqrt3
Cylinder
Volume
V=\pi r^2h
Curved surface
CSA=2\pi rh
Total surface
TSA=2\pi r(r+h)
Hollow cylinder
V=\pi(R^2-r^2)h
Capacity
1\ \text{m}^3=1000\ \ell
Cone
Slant height
\ell=\sqrt{r^2+h^2}
Volume
V=\frac13\pi r^2h
Curved surface
CSA=\pi r\ell
Total surface
TSA=\pi r(\ell+r)
Sphere and hemisphere
Sphere
V=\frac43\pi r^3,\quad S=4\pi r^2
Hemisphere
V=\frac23\pi r^3,\quad S_{\text{curved}}=2\pi r^2,\quad S_{\text{total}}=3\pi r^2
Melting rule
V_{\text{before}}=V_{\text{after}}
Count of small spheres
\text{count}=\left(\frac{R}{r}\right)^3
Prisms, pyramids and painted cubes
Prism volume
V=A_{\text{base}}\times h
Pyramid volume
V=\frac13A_{\text{base}}h
Painted-cube counts
3\text{f}=8,\ 2\text{f}=12(n-2),\ 1\text{f}=6(n-2)^2,\ 0\text{f}=(n-2)^3