Geometry: Figures, Special Triangles, Circles, and Scaling
Core skills
Geometry on the GRE tests relationships rather than measurement. Master: angle rules for parallel lines cut by a transversal; triangle properties (angle sum, isosceles base angles, exterior angle, triangle inequality); the Pythagorean theorem and special right triangles 45–45–90 (sides x, x, x√2) and 30–60–90 (x, x√3, 2x); area and perimeter of standard figures; circles (circumference, area, arcs, sectors, with an inscribed angle equal to half the central angle subtending the same arc); coordinate geometry (slope, distance, midpoint, parallel and perpendicular slopes); and volumes of rectangular solids, cylinders, and spheres.
Traps
- Figures are explicitly not drawn to scale: never infer a right angle, equal length, or parallel line from appearance alone.
- Similar figures scale lengths by k, areas by k², and volumes by k³.
- Area and perimeter are not linked proportionally - doubling a square's perimeter quadruples its area.
- The diagonal of a rectangle is not one of its sides; it comes from the Pythagorean theorem.
- For sectors, the degree measure and the arc length must use the same fraction of the whole circle.
Worked example
Two similar triangles have corresponding sides in the ratio 3:5. If the smaller triangle's area is 18, the larger area is 18 × (5/3)² = 18 × 25/9 = 50. Rule of thumb: write the similarity ratio first, then apply the correct power - square for area, cube for volume - before computing.