A reliable method
Name the requested measure, select a formula, then use units. For triangle side lengths, apply strict triangle inequalities. For similar solids, cube the length scale for volume.
The common trap
A segment can look perpendicular without being perpendicular. Equal perimeters do not force equal areas, and a scale factor for length is not an area multiplier.
Make the connection
Check extreme cases and bounds. A triangle with one side equal to the sum of the others is degenerate; it does not meet the strict triangle condition.
Contrast the cases
Lengths 3,4,7 do not form a nondegenerate triangle because equality at the upper triangle bound is forbidden. Lengths 3,4,6 do. If every length of a similar solid doubles, its surface area scales by four and volume by eight. In a circle of radius 6, a 60° arc has length 2π; a 60° sector instead has area 6π. Both use a sixth of a full figure but different formulas and units.
Transfer and explain
Transfer: for radius 9 and central angle 120°, find arc length 6π and sector area 27π. Explain why an answer in square units cannot be an arc length.
A rectangle has length twice its width and perimeter 36 cm. What is its area in square centimeters?
- If width is w, length is 2w and perimeter is 6w.
- Therefore w = 6, and length = 12.
- Area = 6 × 12 = 72.
Compare the other answer choices
Returns half the perimeter.
Returns the full perimeter.
Treats it as a square of side l.
Forgets the factor 2 in the perimeter formula, doubling both dimensions.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference