2. The Formula Sheet & the Ideas Behind It

You get the formula sheet -- but Level 3 tests whether you can choose and combine the right tools.

Geometry

• Volumes: cylinder \(\pi r^2 h\), cone \(\tfrac{1}{3}\pi r^2 h\), sphere \(\tfrac{4}{3}\pi r^3\), prism/box \(lwh\).
• Surface area (cylinder): \(2\pi r^2 + 2\pi r h\).
• Pythagorean theorem \(a^2+b^2=c^2\); in 3-D the space diagonal is \(\sqrt{l^2+w^2+h^2}\).
• Sector: arc \(= \frac{\theta}{360}\,(2\pi r)\); sector area \(= \frac{\theta}{360}\,(\pi r^2)\).

Coordinate geometry

• Distance \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\); midpoint \(\big(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\big)\); slope \(\tfrac{y_2-y_1}{x_2-x_1}\).

Algebra

• Quadratic formula: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\); vertex at \(t=-\dfrac{b}{2a}\).

Combination ideas the sheet won't state

• Compound growth/decay: multiply by \((1+r)^n\) (decay uses \(1-r\)).
• Similar solids: areas scale by the factor squared, volumes by the factor cubed.
• Average speed for equal distances is the harmonic mean \(\dfrac{2v_1 v_2}{v_1+v_2}\), not the plain average.