A right prism has a regular hexagonal base with area 24 square units and perimeter 18 units. The prism's height is 5 units. What is the total surface area?
A24 + 90 = 114 square units
B2(24) + 18(5) = 138 square units
C18 × 5 = 90 square units
D2(24) + 18 = 66 square units
SA = 2B + Ph = 2(24) + 18(5) = 48 + 90 = 138. The two bases each contribute B = 24, and the lateral surface area is perimeter times height = 18 × 5 = 90. Option A (114) is the classic error of counting only one base instead of two. Option C (90) calculates only the lateral area, forgetting both bases. The formula works identically for any polygon base — hexagon, triangle, or pentagon.
Question 2 Multiple Choice
Why does unfolding a right prism into a net make the lateral surface area calculation straightforward?
AThe net shows that each lateral face is a triangle, making the triangle area formula applicable
BThe lateral faces, when unfolded side by side, form a single rectangle whose width equals the base perimeter and whose height equals the prism height
CThe net eliminates the need to calculate the base area separately
DEach lateral face must be calculated separately even in the net; the net just shows them arranged neatly
Each side of the base polygon corresponds to one lateral rectangular panel. When those panels are unfolded and laid flat side by side, they form one large rectangle. Its height is h (the prism height), and its width is the sum of all base side lengths — the perimeter P. So lateral surface area = P × h. This insight is what makes SA = 2B + Ph not just a formula to memorize but a geometrically transparent result.
Question 3 True / False
The formula SA = 2B + Ph applies to any right prism, regardless of the shape of the base polygon.
TTrue
FFalse
Answer: True
This is the power of deriving the formula from a net rather than memorizing it for specific cases. Whether the base is a triangle, rectangle, regular hexagon, or irregular polygon, the structure is always the same: two congruent bases plus a lateral surface that unfolds into a rectangle of width P and height h. The only thing that changes is how you compute B (base area) and P (base perimeter) for the specific polygon involved.
Question 4 True / False
In the formula SA = 2B + Ph, the variable h represents the height of the base polygon — for example, the height of the triangular base in a triangular prism.
TTrue
FFalse
Answer: False
h is the height of the prism — the perpendicular distance between the two bases, which equals the length of the lateral edges. It has nothing to do with any measurement inside the base polygon. For a triangular prism, the triangle has its own height used in computing B (base area), but that is a separate quantity. Confusing these two heights is a common source of error, especially in triangular prisms where both exist.
Question 5 Short Answer
Explain why the lateral surface area of a right prism equals P × h. Reference what happens when you unfold the lateral faces.
Think about your answer, then reveal below.
Model answer: Each side of the base polygon forms one lateral rectangular face. The width of that rectangle equals the length of that base side; its height equals h, the prism's height. When you cut the lateral faces along their vertical edges and unfold them flat, they lie side by side to form one big rectangle. The total width of this rectangle is the sum of all the base side lengths — the perimeter P. Therefore, the area of the unfolded rectangle is P × h, which is the entire lateral surface area.
This reasoning makes the formula understandable rather than arbitrary. It also generalizes immediately to any right prism: the net always produces two congruent bases and one lateral rectangle, regardless of the base polygon. Students who understand the net derivation can reconstruct the formula rather than needing to memorize it.