Decision and conditions
Separate a boundary length, a planar area and the surface or volume of a solid. All radii and heights use the same unit. A closed cylinder has two circular bases.
Worked model
A sector of radius 18 has central angle 120°. Its area is cπ. What is c? 120° is one third of a complete turn. Full area is 324π. The sector coefficient is 324/3 = 108. Before calculating: Take the angle’s fraction of the full circle area, not the circumference.
Contrast: a different decision
Two concentric circles have radii 3 and 5. The area between them is cπ. What is c? Outer area coefficient is 25. Inner area coefficient is 9. The difference is 16. Before calculating: Subtract the two disk areas; do not square the difference of the radii.
Transfer with solution
A closed cylinder has radius 3 and height 2. Its total surface area is cπ. What is c? The two bases have combined area coefficient 18. The curved surface has coefficient 2rh = 12. The total coefficient is 30. Before calculating: A closed cylinder includes a curved wall and two circular bases.
Check the boundary
Subtracting radii before squaring does not compute an annulus area. A rectangular box has perpendicular edge lengths 3, 6, and 6. What is its space diagonal? The squared diagonal is 9+36+36 = 81. Take the positive square root. The diagonal is 9.
A sector of radius 18 has central angle 120°. Its area is cπ. What is c?
- 120° is one third of a complete turn.
- Full area is 324π.
- The sector coefficient is 324/3 = 108.
Compare the other answer choices
Uses one third of the circumference: 2(18)/3 = 12 as the π coefficient.
Divides the radius by three before squaring.
Uses the complementary 240° sector.
Uses the entire circle instead of the sector.
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 overview