Geometry and Measurement: Triangles, Circles, Solids and Coordinates
What you must understand
This domain covers planar figures (triangles, quadrilaterals, circles) and solids (prisms, pyramids, cylinders, cones, spheres), plus coordinate geometry. You need perimeter, area, surface area, volume, the Pythagorean theorem, similarity, trigonometric ratios in right triangles, and the distance and midpoint formulas. Angle facts (sum of triangle angles, inscribed angle theorem) are frequently tested.
Traps that appear in questions
- Units. Mixing cm and m, or reporting area in linear units, is a common slip. Convert first, compute second.
- Radius vs diameter. Circle area uses (r); a question may give the diameter.
- Similar figures. Lengths scale by (k), areas by (k^2), volumes by (k^3). A doubling of the side quadruples the area and octuples the volume.
- Right-triangle ratios. (\sin = \text{opposite}/\text{hypotenuse}), (\cos = \text{adjacent}/\text{hypotenuse}), (\tan = \text{opposite}/\text{adjacent}). Mixing opposite and adjacent is the classic error.
- Inscribed angle. An inscribed angle is half the central angle subtending the same arc.
Worked example
A right circular cone has radius (r=3) cm and height (h=4) cm. Find its volume and slant height. Slant height (l=\sqrt{r^2+h^2}=\sqrt{9+16}=5) cm. Volume (V=\tfrac13\pi r^2 h=\tfrac13\pi\cdot 9\cdot 4=12\pi) cm³.
Rule of thumb: sketch the figure, label every known quantity with its unit, and identify which formula applies before substituting numbers.