8.3 Unit Conversions and Basic Geometry

Metric Length Conversions

Nearly all measurement questions in Police Problem Solving use the metric system, which makes conversions a matter of shifting a decimal point rather than multiplying by an awkward number. The three units that come up most often are centimeters, meters, and kilometers, and the relationship between each pair is always a clean power of ten.

The pattern to remember is simple: converting from a smaller unit to a larger unit (cm to m, or m to km) always divides, while converting from a larger unit to a smaller unit (km to m, or m to cm) always multiplies. Both directions use the exact same relationship number - only whether you multiply or divide changes.

Worked example: an officer measures a skid mark at a collision scene as 750 cm long. To report the length in meters, divide by 100: 750 divided by 100 equals 7.5 m. A second worked example converts a patrol route logged at 3,200 m into kilometers by dividing by 1,000: 3,200 divided by 1,000 equals 3.2 km. Reversing that same route length from 2.4 km back into meters multiplies instead of dividing: 2.4 x 1,000 equals 2,400 m.

The identical logic applies to converting minutes and hours, which is why it belongs in the same mental toolkit as metric conversions even though the underlying base number is 60 instead of 10: dividing minutes by 60 converts them into hours (the larger unit), and multiplying hours by 60 converts them back into minutes (the smaller unit).

Rectangle Perimeter and Area

Two rectangle formulas get tested constantly, and confusing them is the most common geometry mistake on a timed, no-calculator test: perimeter, the distance all the way around a shape's outer edge, and area, the amount of flat surface the shape covers.

A useful way to keep the two straight in a policing context: perimeter is how much tape you would need to cordon off a scene, and area is how much ground your search team actually has to cover inside that tape. Worked example: a roughly rectangular crime scene measures 18 m long and 12 m wide. The tape needed for the perimeter is 2 x (18 plus 12), which is 2 x 30, or 60 m. The ground area investigators must search is 18 x 12, or 216 square meters. Notice that the two answers use entirely different formulas and entirely different units (meters versus square meters) even though they describe the exact same rectangle - which is exactly why exam distractors often list one shape's perimeter as though it were its area, or vice versa.

Triangles, Circles, and Volume Basics

Beyond rectangles, three additional shapes come up in everyday policing scenarios often enough to be worth memorizing cold: triangles, circles, and simple rectangular volumes.

For circle problems on a no-calculator section, using pi as approximately 22/7 instead of 3.14 often produces a whole number, provided the radius is a multiple of 7 - a pattern worth watching for, because it turns an otherwise messy decimal multiplication into simple whole-number arithmetic.

Worked example, triangle: a roughly triangular parking area used as a staging zone has a 16 m base and a 9 m height. Area equals one-half x 16 x 9. Multiply first: 16 x 9 equals 144, then take half: 144 divided by 2 equals 72. The staging area covers 72 square meters.

Worked example, circle: a circular fountain in a public square has a radius of 7 m. Because 7 is exactly the denominator in 22/7, using that fraction for pi keeps the arithmetic clean. Circumference equals 2 x 22/7 x 7. The 7 in the denominator cancels directly against the 7 m radius, leaving 2 x 22, or 44 m of circumference. Area equals 22/7 x 7 x 7, and again one 7 cancels against the denominator, leaving 22 x 7, or 154 square meters.

Worked example, volume: an evidence storage container measures 2 m long, 1.5 m wide, and 1 m high. Volume equals length x width x height, or 2 x 1.5 x 1. Multiply in two steps: 2 x 1.5 equals 3, then 3 x 1 equals 3. The container holds 3 cubic meters.

Across all three shapes, the practical exam strategy is identical: write down which formula the question is actually asking for before doing any arithmetic, substitute the given numbers directly into that formula, and look for a cancellation or a round number - like a radius divisible by 7, or dimensions that multiply evenly - that turns the calculation into something manageable entirely by hand within the time available.