Geometry, Measurement, and Data Analysis
Geometry
Angles, triangles, and three-dimensional figures
Angle relationships anchor many problems: a straight line measures 180°, a full circle 360°, vertical angles are equal, complementary angles sum to 90°, and supplementary angles sum to 180°. The angles of any triangle sum to 180°, and the longest side faces the largest angle.
The Pythagorean theorem, a² + b² = c², applies only to right triangles, where c is the hypotenuse opposite the right angle. Special right triangles follow fixed ratios: 45-45-90 sides are in ratio 1 : 1 : √2, and 30-60-90 sides in ratio 1 : √3 : 2. Similar triangles have equal angles and proportional sides, so set up a proportion and cross-multiply. Congruent figures are similar with a scale factor of 1.
Area formulas: rectangle A = lw, triangle A = ½bh with height perpendicular to the base, circle C = 2πr and A = πr². Volume: prisms and cylinders V = Bh, cones and pyramids V = ⅓Bh, spheres V = (4/3)πr³.
Traps: using diameter where radius is required, mixing perimeter units with area units, substituting slant height for perpendicular height, and applying the Pythagorean theorem to non-right triangles.
Worked example: a right triangle with legs 6 and 8 has hypotenuse √(36 + 64) = √100 = 10, and area ½(6)(8) = 24 square units.
Measurement and Data Analysis
Convert with care, summarize with sense
Unit conversions are performed by multiplying by fractions equal to 1, arranged so unwanted units cancel: 3.5 hours × (60 minutes / 1 hour) = 210 minutes. Area and volume conversions square or cube the linear factor, so 1 m² = 10,000 cm², not 100 cm². Rates combine units, such as miles per gallon or cost per ounce.
Descriptive statistics: the mean is the sum divided by the count; the median is the middle value of the ordered data, or the average of the two middle values when the count is even; the mode is the most frequent value; the range is maximum minus minimum. The mean responds strongly to outliers while the median resists them, so compare both when data is skewed. A weighted average multiplies each value by its weight, sums, and divides by the total weight.
Probability equals favorable outcomes divided by total outcomes, and the complement of an event is 1 minus its probability. Percent change is (new − old) / old × 100; note that a 20% increase followed by a 20% decrease does not return to the starting value.
Traps: finding the median without ordering the data first, misreading a graph's axis scale, converting only one dimension of an area, and confusing "percent of" with "percent change."
Worked example: for the scores 70, 80, 90, 90, the mean is 82.5, the median is 85, the mode is 90, and the range is 20.