Mensuration: Surface Area & Volume

At a glance 11 formulas

1. 1Cuboid$\text{volume } lbh,\ \text{surface } 2(lb + bh + hl)$$l, b, h = \text{edges};\ \text{longest rod } \sqrt{l^2 + b^2 + h^2}$
2. 2Cylinder, radius r$\text{volume} = \pi r^2 h,\ \text{curved} = 2\pi r h$$h = \text{height; total surface} = 2\pi r(r + h)$
3. 3Cone, slant height l$\text{volume} = \tfrac{1}{3}\pi r^2 h,\ \text{curved} = \pi r l$$r, h = \text{radius, height};\ l = \sqrt{r^2 + h^2}$
4. 4Sphere, radius r$\text{volume} = \tfrac{4}{3}\pi r^3,\ \text{surface} = 4\pi r^2$$\text{hemisphere: } \tfrac{2}{3}\pi r^3,\ \text{total surface } 3\pi r^2$
5. 5Units and flow$1\ \text{m}^3 = 1000\ \text{L},\ 1\ \text{L} = 1000\ \text{cm}^3$$\text{pipe flow} = \pi r^2 \times \text{speed},\ r = \text{pipe radius}$
6. 6Melting and recastingVolume stays the same and surface area does not: pieces = old volume ÷ volume of one new piece. Cancel $\pi$ and common factors before multiplying.
7. 7Water displaced$\text{rise} = \dfrac{\text{solid's volume}}{\text{vessel's base area}}$
8. 8Box from three faces$\text{volume} = \sqrt{xyz}$$x, y, z = \text{face areas at one corner}$
9. 9Rolling a sheetThe edge that goes round becomes the circumference $2\pi r$ and the other edge the height. Both ways the curved surface is the whole sheet, but the volumes differ.
10. 10Sector into a cone$r = \dfrac{\theta}{360^\circ} \times R,\ l = R$$R, \theta = \text{sector};\ r = \text{base radius},\ l = \text{slant}$
11. 11Similar cones$\dfrac{\text{small volume}}{\text{big volume}} = k^3$$k = \text{height ratio};\ \text{frustum} = 1 - k^3 \text{ of it}$