Geometry and Measurement
Geometry and Measurement
This domain covers Euclidean and coordinate geometry, congruence and similarity, circles, right-triangle and general trigonometry, transformations, and measurement of length, area, and volume in two and three dimensions. Questions often combine a figure's algebraic description with its geometric meaning.
Core rules and concepts
- Circle equation: (x − h)² + (y − k)² = r², center (h, k), radius r. Complete the square to convert from general form.
- Similar figures: corresponding lengths scale by k, areas by k², volumes by k³.
- Trigonometry: SOH-CAH-TOA for right triangles; Law of Sines a/sin A = b/sin B; Law of Cosines c² = a² + b² − 2ab cos C.
- Radians: arc length s = rθ and sector area A = ½r²θ when θ is in radians.
- Volume formulas: prism/cylinder V = Bh; pyramid/cone V = ⅓Bh; sphere V = (4/3)πr³.
Worked example
Find the center and radius of x² + y² − 6x + 4y − 12 = 0. Group: (x² − 6x) + (y² + 4y) = 12. Complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4. So (x − 3)² + (y + 2)² = 25. Center (3, −2), radius 5. Reasoning: adding 9 and 4 to the left requires adding them to the right to preserve equality.
Traps that cost marks
- Forgetting to add the completed-square constants to both sides.
- Using degrees in s = rθ or A = ½r²θ.
- Scaling area by k instead of k² in similar figures.
- Sign errors: (y + 2)² has center y = −2, not +2.