棱柱léngzhù

Core Concept

A prism (棱柱) is a polyhedron with two congruent, parallel faces (the bases, 底面) joined by parallelogram lateral faces (侧面).

The Defining Conditions

A solid is a prism if and only if both hold:

1. Two faces are congruent polygons lying in parallel planes.
2. All remaining faces are parallelograms.

CSCA questions phrased as 判断几何体是否为棱柱 test exactly this. A common counterexample: a solid with two parallel similar but not congruent bases is a frustum (棱台), not a prism — its lateral faces are trapezoids rather than parallelograms.

Classification

Type	Chinese	Condition
Oblique prism	斜棱柱	lateral edges not perpendicular to base
Right prism	直棱柱	lateral edges perpendicular to base
Regular prism	正棱柱	right prism and regular polygon base

A cuboid (长方体) is a right prism with rectangular bases; a cube (正方体) is the special case with all edges equal.

Core Formulas

$V = S_{\text{base}} \cdot h$

$S_{\text{lateral}} = c \cdot l \quad (\text{right prism: } c = \text{base perimeter}, \ l = \text{lateral edge})$

$S_{\text{total}} = S_{\text{lateral}} + 2S_{\text{base}}$

Note that $h$ is the perpendicular distance between the two bases. For an oblique prism this is not the lateral edge length — a distinction the exam exploits.

Key Properties

Property 1: Uniform Cross-Section

Every cross-section parallel to the base is congruent to the base. This is why the volume is simply "base area × height" — the solid has the same cross-section all the way up.

Property 2: Space Diagonal of a Cuboid

For a cuboid with edges $a$, $b$, $c$:

$d = \sqrt{a^2 + b^2 + c^2}$

This is the 3-D distance formula, and it links directly to circumscribed-sphere problems (the space diagonal is the sphere's diameter).

Property 3: Counting Elements

An $n$-gonal prism has:

• $2n$ vertices
• $3n$ edges ($2n$ base edges + $n$ lateral edges)
• $n + 2$ faces

Check with Euler's formula: $V - E + F = 2n - 3n + (n+2) = 2$ ✓

CSCA Practice Problems

💡 Note: These problems follow the CSCA exam syllabus and Chinese standardized test formats.

Example 1: Basic (Difficulty ★★☆☆☆)

A right prism has a square base of side $3$ and height $6$. Its total surface area is:

Options:

• A. $54$
• B. $72$
• C. $90$
• D. $108$

Detailed Solution:

1. Base area: $S_{\text{base}} = 3^2 = 9$

2. Base perimeter: $c = 4 \times 3 = 12$

3. Lateral surface area: $S_{\text{lateral}} = c \cdot h = 12 \times 6 = 72$

4. Total surface area: $S_{\text{total}} = 72 + 2(9) = 90$

Option B is the trap for anyone who stops at the lateral area and forgets the two bases.

Answer: C

---

Example 2: Intermediate (Difficulty ★★★☆☆)

A cuboid has edges $3$, $4$, and $12$. Find the radius of its circumscribed sphere.

Options:

• A. $\frac{13}{2}$
• B. $13$
• C. $\frac{\sqrt{13}}{2}$
• D. $6$

Detailed Solution:

1. The circumscribed sphere passes through all eight vertices, so its diameter equals the space diagonal: $2R = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$

2. Therefore: $R = \frac{13}{2}$

Option B is the trap for reporting the diameter as the radius.

Note: $3$-$4$-$12$-$13$ is a Pythagorean quadruple, one of a small set that CSCA reuses — worth recognising alongside $1$-$2$-$2$-$3$ and $2$-$3$-$6$-$7$.

Answer: A

Common Mistakes

❌ Mistake 1: Using the lateral edge as the height in an oblique prism

In a 斜棱柱, the lateral edge is longer than the perpendicular height. $V = S_{\text{base}} \cdot h$ requires the perpendicular height.

❌ Mistake 2: Forgetting the two bases in total surface area

$S_{\text{total}} = S_{\text{lateral}} + 2S_{\text{base}}$ — the factor of $2$ is dropped surprisingly often.

❌ Mistake 3: Calling a frustum a prism

If the two parallel faces are not congruent, it is a 棱台, not a 棱柱.

❌ Mistake 4: Confusing 正棱柱 with 直棱柱

A right prism (直棱柱) only requires perpendicular lateral edges. A regular prism (正棱柱) additionally requires a regular polygon base. Every 正棱柱 is a 直棱柱, but not conversely.

Study Tips

1. ✅ Memorise the two defining conditions — they answer all 判断 questions directly
2. ✅ Distinguish perpendicular height from lateral edge before applying the volume formula
3. ✅ Learn the space-diagonal formula — it bridges prisms and circumscribed spheres
4. ✅ Use Euler's formula $V - E + F = 2$ to check your element counts
5. ✅ Draw the net (展开图) when computing surface area — it makes the two bases impossible to forget

---

💡 Exam Tip: 棱柱表面积的有关计算 and 判断几何体是否为棱柱 both appear in CSCA solid geometry. Prisms also serve as the setup for circumscribed-sphere questions, so the space-diagonal formula earns marks in two topics at once.