Geometry: Lines, Triangles, Circles, and Solids
What you must know
Geometry questions center on angle relationships, triangle properties (angle sum, isosceles and equilateral cases, Pythagorean theorem and common triples like 3-4-5 and 5-12-13, special right triangles 45-45-90 and 30-60-90), circle formulas (circumference, area, arcs, sectors), coordinate geometry (slope, distance, midpoint), and volume and surface area of rectangular solids, cylinders, and spheres.
Traps
- Assuming a figure is drawn to scale - diagrams may mislead; rely on given measures and stated relationships.
- Confusing radius and diameter in area and circumference formulas.
- Forgetting that the largest angle faces the longest side, and that a triangle's third side must lie strictly between the difference and the sum of the other two.
- Mixing up area and perimeter when a problem asks about fencing versus covering.
- Slope of perpendicular lines: it is the negative reciprocal, not just the negative.
Worked example
A right triangle has legs 6 and 8. The hypotenuse is √(36 + 64) = √100 = 10. If a problem instead gives the hypotenuse 13 and one leg 5, the other leg is √(169 − 25) = √144 = 12. Recognizing the 5-12-13 triple lets you skip the arithmetic entirely.
Rule of thumb
Memorize the common Pythagorean triples and the two special right-triangle ratios. For any solid, identify whether the question asks for volume (cubic units) or surface area (square units) before computing.