Mensuration (3D)
🔒 Log in to trackmedium importance~2 Q in Tier 121 formulas⚡ 15 shortcuts5 subtopics
Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.
Cube and cuboid
Cuboid
V=lbh,\quad \text{LSA}=2h(l+b),\quad \text{TSA}=2(lb+bh+hl)
l, b, h are the three dimensions.
Cube
V=a^3,\quad \text{LSA}=4a^2,\quad \text{TSA}=6a^2
a = edge.
Diagonals
d=\sqrt{l^2+b^2+h^2},\quad d_{\text{cube}}=a\sqrt3
Corner to opposite corner.
Capacity
1\text{ m}^3=1000\text{ L},\quad 1\text{ L}=1000\text{ cm}^3
Convert once, at the end.
Painted cube counts
8,\ 12(n-2),\ 6(n-2)^2,\ (n-2)^3
3, 2, 1, 0 painted faces for n by n by n.
Cylinder
Cylinder
V=\pi r^2h,\quad \text{CSA}=2\pi rh,\quad \text{TSA}=2\pi r(h+r)
r = radius, h = height.
Missing height
h=\frac{V}{\pi r^2}
Reverse of the volume formula.
Capacity
\text{litres}=\text{m}^3\times1000
Same conversion as tanks.
Dimension change
V\propto r^2h,\quad \text{CSA}\propto rh
Scale each symbol by its own factor.
Cone
Slant height
\ell=\sqrt{r^2+h^2}
Radius, height, slant: a right triangle.
Cone volume
V=\frac{1}{3}\pi r^2h
One-third of the same-base cylinder.
Cone surfaces
\text{CSA}=\pi r\ell,\quad \text{TSA}=\pi r(\ell+r)
Skirt alone, or skirt plus base.
Same base ratios
V_{\text{cone}}=\frac{V_{\text{cyl}}}{3}
Equal base and height.
Sphere and hemisphere
Sphere
V=\frac{4}{3}\pi r^3,\quad S=4\pi r^2
One radius drives both.
Hemisphere
V=\frac{2}{3}\pi r^3,\quad \text{curved}=2\pi r^2,\quad \text{total}=3\pi r^2
Total adds the flat circle.
Melting into n parts
r_{\text{small}}^3=\frac{R^3}{n}
Divide the cubed length, then cube-root.
Radius scaling
V\to k^3V,\quad S\to k^2S
k = radius scale factor.
Prisms, pyramids and painted cubes
Prism
V=\text{base area}\times\text{length}
Base can be any polygon.
Pyramid
V=\frac{1}{3}\times\text{base area}\times h
h = perpendicular height.
Frustum
V=\frac{\pi h}{3}(R^2+r^2+Rr)
R, r = the two end radii.
Lateral surfaces
\text{prism}=P\times L,\quad \text{pyramid}=\frac12 P\times\ell
P = base perimeter; L = length; slant for pyramid.