Engineering Mathematics and Applied Calculations
This domain checks calculation fluency applied to workshop and design problems: SI units and prefixes, areas and volumes, Pythagoras and trigonometry, ratio and gear trains, taper, percentage and sensible rounding.
Foundations
- Prefixes: k = 10³, m = 10⁻³, µ = 10⁻⁶. Convert to base units before calculating.
- Right triangle: a² + b² = c²; sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
- Circle: area = πd²/4, circumference = πd.
- Gear train: speed ratio = driver teeth ÷ driven teeth; a 20-tooth driver turning 1400 rpm drives a 70-tooth gear at 1400 × 20/70 = 400 rpm, with torque rising by the same 3.5 factor.
Taper and angle
Taper = (D − d)/L, expressed as 1:n. The half-angle uses the radius difference: tan α = (D − d)/(2L).
Worked example. A taper has D = 40 mm, d = 34 mm and L = 60 mm. Taper = 6/60 = 1:10. Half-angle: tan α = 6/(2 × 60) = 0.05, so α ≈ 2.86°, and the included angle is about 5.72°. Set the compound slide to the half-angle, not the included angle.
Traps. Using the diameter difference instead of the radius difference in the angle formula; inverting a gear ratio; leaving the calculator in radian mode; rounding mid-calculation and then rounding again; mixing millimetres and metres in a stress or speed formula.