Choose a representation
List the contributing pieces before combining formulas. For similar solids, use a cube of the length scale. A cone occupies one third of the cylinder with equal base and height.
Worked example: follow the quantity
A closed cylinder has radius 3 cm and height 5 cm. Its total surface area is Kπ cm². What is K? Include two circular ends and the curved rectangular surface. Add 2 r² to 2 rh, keeping π symbolic. Substitute the given numbers: 2 × 3^2+2 × 3 × 5 = 48. Therefore the requested value is 48.
Contrast a plausible mistake
For that example, the tempting answer 18 follows this error: Include the two ends but omit the curved surface. The requested quantity is 48. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
Two similar spheres have radius ratio 1:3. The smaller volume is 5 cm³. What is the larger volume? Volume uses three length dimensions. Multiply the small volume by the cube of the radius ratio. Substitute the given numbers: 5 × 3^3 = 135. Therefore the requested value is 135.
Transfer before checking
A closed cylinder has radius 4 cm and height 6 cm. Its total surface area is Kπ cm². What is K? Try without viewing the answer. Check: 80. Include two circular ends and the curved rectangular surface. Add 2 r² to 2 rh, keeping π symbolic. Substitute the given numbers: 2 × 4^2+2 × 4 × 6 = 80.
Scope and conditions
A drawing does not tell you which faces are exposed. Check closed versus open containers and distinguish a removed corner from an added strip. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
A closed cylinder has radius 3 cm and height 5 cm. Its total surface area is Kπ cm². What is K?
- Include two circular ends and the curved rectangular surface.
- Add 2 r² to 2 rh, keeping π symbolic.
- Substitute the given numbers: 2 × 3^2+2 × 3 × 5 = 48.
- Therefore the requested value is 48.
Compare the other answer choices
Include the two ends but omit the curved surface.
Include the curved surface but omit both ends.
Include only one circular end.
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.
ACT Mathematics content