Choose a representation
Use the stated volume formula with consistent units. Recover height by dividing volume by its base factor. For a rectangular box, combine the squares of all three edge lengths.
Worked example
A cylinder has radius 4 and height 6. Using V=πr²h, its volume is Kπ. What is K? Square the radius to get the coefficient of its circular base area. Multiply that base coefficient by the cylinder height. Substitute the given numbers: 4^2 × 6 = 96. The requested value is 96.
Contrast a wrong operation
In that example, 24 follows this error: Use radius rather than radius squared. The correct requested value is 96. Explain which quantity the mistaken method is actually calculating.
Connect a second form
A rectangular box has perpendicular edge lengths 12,16,48. What is the length of its space diagonal? The base diagonal has squared length equal to the sum of the first two edge squares. Include the third edge square and take the positive square root. d² = 12² + 16² + 48² = 2704; d = 52. The requested value is 52.
Transfer and check
A cylinder has radius 5 and height 7. Using V=πr²h, its volume is Kπ. What is K? Pause before checking. Answer: 175. Square the radius to get the coefficient of its circular base area. Multiply that base coefficient by the cylinder height. Substitute the given numbers: 5^2 × 7 = 175.
Boundaries of the method
A cone needs the factor one third. A box diagonal is not the sum of edge lengths. Complementary-angle identities apply to the two acute angles of a right triangle. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
A cylinder has radius 4 and height 6. Using V=πr²h, its volume is Kπ. What is K?
- Square the radius to get the coefficient of its circular base area.
- Multiply that base coefficient by the cylinder height.
- Substitute the given numbers: 4^2 × 6 = 96.
- The requested value is 96.
Compare the other answer choices
Use radius rather than radius squared.
Apply a cone factor to a cylinder.
Compute total surface-area coefficient instead of volume.
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.
Math specifications