The decision and its conditions
Match a formula to the quantity being measured. Circumference follows a boundary and area covers a surface. A diameter must be halved before entering a radius-squared formula. For a trapezoid, the height is perpendicular to the parallel bases, and averaging those bases explains the area rule. A ring is the area of an outer circle with an inner circle removed; subtract the squared radii rather than squaring the width of the ring. Exact answers can retain π as a factor when the prompt asks for its coefficient.
Worked example
A circular lid has circumference 10π centimeters. Using C=2πr, find its radius in centimeters. Circumference is a boundary length, not an area. Cancel π from 10π=2πr, leaving 10=2r. Dividing by two gives radius 5 cm.
A contrasting decision
A circular mat has diameter 8 feet. Its area is kπ square feet. Find k. Radius is half the diameter: 8/2=4 feet. Area is π times radius squared. The coefficient k is 4²=16.
Diagnose a competing method
The first example’s tempting answer 10 follows this mistake: Report the diameter rather than the radius. Use the supplied circumference rule and isolate the radius.
Transfer before revealing
A trapezoid has parallel bases 6 m and 12 m, and perpendicular height 4 m. Using A=(b₁+b₂)h/2, what is its area? Pause and identify the requested quantity. Solution: Average the parallel bases: (6+12)/2=9. Multiply by perpendicular height 4. The area is 36 m².
Check the boundary of the method
A ring is the area of an outer circle with an inner circle removed; subtract the squared radii rather than squaring the width of the ring. Convert diameter to radius before squaring. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
A circular lid has circumference 10π centimeters. Using C=2πr, find its radius in centimeters.
- Circumference is a boundary length, not an area.
- Cancel π from 10π=2πr, leaving 10=2r.
- Dividing by two gives radius 5 cm.
Compare the other answer choices
Report the diameter rather than the radius.
Square the circumference coefficient instead of solving the linear formula.
Divide by two twice.
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.
GED Mathematical Reasoning Assessment Guide for Educators