Pythagoras, mensuration and 3D

Formulas to know (M5.7, M5.16)

• Pythagoras \(a^2 + b^2 = c^2\); cuboid diagonal \(\sqrt{a^2 + b^2 + c^2}\); pyramid height from a slant edge via half the base diagonal.
• Triangle \(\frac12 bh\); parallelogram \(bh\); trapezium \(\frac12(a + b)h\); circle \(\pi r^2\), circumference \(2\pi r\); sector \(\frac{\theta}{360}\pi r^2\).
• Prism volume = cross-section \(\times\) length; cylinder \(\pi r^2 h\), curved surface \(2\pi rh\), total \(2\pi r^2 + 2\pi rh\).
• Given if needed: cone \(\frac13\pi r^2h\) (curved surface \(\pi rl\)), sphere \(\frac43\pi r^3\) (surface \(4\pi r^2\)), pyramid \(\frac13 \times\) base \(\times\) height.

Trigonometry in right-angled triangles (M5.17, M5.18)

• \(\sin = \frac{\text{opp}}{\text{hyp}}\), \(\cos = \frac{\text{adj}}{\text{hyp}}\), \(\tan = \frac{\text{opp}}{\text{adj}}\); exact values for \(30^\circ, 45^\circ, 60^\circ\) from the \(1, 1, \sqrt2\) and \(1, \sqrt3, 2\) triangles.
• Angles of elevation/depression are measured from the horizontal; the angle of depression from the top equals the angle of elevation from the bottom.
• Similar solids: if lengths scale by \(k\), areas scale by \(k^2\) and volumes by \(k^3\).
• Plans and elevations: reason about the minimum and maximum number of cubes consistent with all three views.