Geometry and Measurement
Geometry and Measurement
This domain tests spatial reasoning and measurement formulas: angle relationships, triangle congruence and similarity, perimeter/area/volume, coordinate geometry, circles, and unit conversion. Many items are multi-step and require choosing the right formula before computing.
Core rules
- Angles on a line sum to 180°; angles around a point sum to 360°. Vertical angles are equal. When parallel lines are cut by a transversal, alternate interior and corresponding angles are equal.
- The angles of a triangle sum to 180°. The longest side is opposite the largest angle. The Pythagorean theorem applies only to right triangles: a^2 + b^2 = c^2, where c is the hypotenuse.
- Similar figures have equal corresponding angles and proportional sides. If the linear scale factor is k, areas scale by k^2 and volumes by k^3.
- Circle formulas: circumference C = 2πr, area A = πr^2, arc length = (θ/360)(2πr), sector area = (θ/360)(πr^2).
- Volume: prism/cylinder V = Bh; pyramid/cone V = (1/3)Bh; sphere V = (4/3)πr^3. Surface area is the sum of face areas.
- Convert units by multiplying by conversion factors written as fractions so unwanted units cancel.
Worked example
Two similar cylinders have radii in the ratio 2:3. The smaller volume is 40 cm^3. The linear scale factor from small to large is 3/2, so the volume factor is (3/2)^3 = 27/8. Thus the larger volume is 40 × 27/8 = 5 × 27 = 135 cm^3. Using the radius ratio directly, the volume ratio is 2^3:3^3 = 8:27, and 40/8 × 27 = 135.
Traps
Do not use a radius where a diameter is given, or forget to square the radius in area. A scale factor k changes area by k^2 and volume by k^3-not k. The Pythagorean theorem cannot be applied to non-right triangles. In unit conversions, check that the final unit is the one requested; cm^2 to m^2 uses 10,000, not 100.