Geometry and Measurement: Perimeter, Area, Volume, and Unit Conversion
Core skills
Perimeter is the distance around a figure, area is the number of square units covering it, and volume is the number of cubic units filling a solid. Know the standard formulas: rectangle A = lw and P = 2l + 2w; triangle A = 1/2 bh, where the height must be perpendicular to the base; circle A = pi r squared and C = 2 pi r; trapezoid A = 1/2 (b1 + b2)h; rectangular prism V = lwh; cylinder V = pi r squared h. Convert units before calculating whenever measurements are mixed, and square or cube the conversion factor when converting area or volume (1 ft = 12 in, but 1 square foot = 144 square inches and 1 cubic foot = 1728 cubic inches). Metric prefixes shift the decimal: kilo-, centi-, milli-. Also expect complementary angles (sum 90 degrees), supplementary angles (sum 180 degrees), and right-triangle problems solved with a squared + b squared = c squared.
Traps
Confusing radius with diameter; omitting the 1/2 in triangle and trapezoid formulas; reporting area in linear units or volume in square units; using a slanted side as the height of a triangle; converting only one dimension of a figure; and adding lengths written in different units without converting first.
Worked example
A cylindrical can has radius 3 cm and height 10 cm. Volume = pi r squared h = pi(3 squared)(10) = 90 pi, about 283 cubic centimeters. Rule of thumb: write the formula, substitute with units, then compute, and label the result in square units for area and cubic units for volume.