A cone has vertical height 4 and base radius 3. What is its lateral surface area?
A12π — computed using the vertical height in place of the slant height
B15π — computed using the correct slant height of 5
C9π — the base area only, omitting the lateral face
D25π — computed by adding height and radius before multiplying
The lateral area of a cone is πrl, where l is the slant height, not the vertical height. First compute slant height using the Pythagorean theorem: l² = h² + r² = 16 + 9 = 25, so l = 5. Lateral area = π(3)(5) = 15π. Option A is the most common error — plugging the vertical height (4) directly into the formula without computing slant height first.
Question 2 Multiple Choice
When the lateral surface of a cone is cut along one edge and unrolled flat, what shape results?
AA rectangle, because the surface wraps smoothly around a circular base
BA triangle, because the cone tapers to a point
CA sector of a circle, with the slant height as its radius
DA full circle with the base radius as its radius
Cutting the lateral surface of a cone along a straight line from apex to base edge and unrolling it produces a flat sector (pie slice) of a larger circle. The radius of that sector is the slant height l, and the arc length of the sector equals the base circumference 2πr. Working out the area of this sector yields the lateral area formula πrl. This 'net' approach is why the formula works.
Question 3 True / False
The slant height of a right cone equals its vertical height.
TTrue
FFalse
Answer: False
The vertical height h drops perpendicularly from the apex to the center of the base. The slant height l runs from the apex diagonally to the rim of the base. Together with the radius r, they form a right triangle where l is the hypotenuse: l² = h² + r². Since r > 0 for any real cone, l is always strictly greater than h.
Question 4 True / False
The slant height of a cone is always greater than its vertical height.
TTrue
FFalse
Answer: True
In the right triangle formed by the vertical height h (vertical leg), radius r (horizontal leg), and slant height l (hypotenuse), the hypotenuse is always longer than either leg when both legs are positive. Since any real cone has r > 0, we have l = √(h² + r²) > h. This is why substituting h for l always underestimates the lateral area.
Question 5 Short Answer
Why must you compute slant height before applying the cone or pyramid surface area formula, even when the problem gives you the vertical height?
Think about your answer, then reveal below.
Model answer: The lateral area formulas (πrl for cones, ½Pl for pyramids) come from the area of the actual slanted faces — the surface you would walk on going up the side. The slant height is the distance from apex to base edge measured along that face. Vertical height is perpendicular to the base and lies entirely inside the solid; it never appears on any lateral face. The Pythagorean theorem l² = h² + r² converts the given vertical height and radius into the slant height the formula needs.
This is the most common error in these problems. Students see h in the problem and plug it into the formula as if it were l. The fix is to treat every problem as two steps: (1) find l using the Pythagorean theorem, (2) substitute l into the area formula. The error is not algebraic — it is conceptual: confusing a height that points straight up with a height that runs up a slanted face.