棱锥léngzhuī

Core Concept

A pyramid (棱锥) is a polyhedron formed by a polygon base (底面) and a point not in the base plane (the apex, 顶点), with triangular lateral faces joining the apex to every base edge.

A pyramid with an $n$-gonal base is called an $n$-gonal pyramid; the triangular case (四面体, tetrahedron) has four faces in total and is the most common CSCA setting.

The Volume Formula

$V = \frac{1}{3} S_{\text{base}} \cdot h$

The $\frac{1}{3}$ is the whole point of this topic. A pyramid holds exactly one third of the prism with the same base and height. Every year, students lose marks by writing $S_{\text{base}} \cdot h$ out of habit from prisms.

Here $h$ is the perpendicular distance from the apex to the base plane — not the slant height, and not a lateral edge.

Regular Pyramids (正棱锥)

A pyramid is regular when:

1. The base is a regular polygon, and
2. The apex lies directly above the centre of the base.

Regular pyramids have two extra useful quantities:

• Slant height (斜高) $l'$ — the height of a lateral face, measured from the apex to the midpoint of a base edge
• Lateral edge (侧棱) $l$ — from the apex to a base vertex

For a regular pyramid:

$S_{\text{lateral}} = \frac{1}{2} c \cdot l' \qquad (c = \text{base perimeter})$

The Three Right Triangles

Nearly every regular-pyramid computation reduces to one of these three right triangles. Let $r$ be the apothem of the base (centre to edge midpoint) and $R$ the circumradius (centre to vertex):

Triangle	Legs	Hypotenuse	Relation
height–apothem–slant	$h$, $r$	$l'$	$l'^2 = h^2 + r^2$
height–circumradius–lateral edge	$h$, $R$	$l$	$l^2 = h^2 + R^2$
slant–half edge–lateral edge	$l'$, $\frac{a}{2}$	$l$	$l^2 = l'^2 + \frac{a^2}{4}$

Drawing the correct one is usually the entire difficulty of a CSCA pyramid problem.

CSCA Practice Problems

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

Example 1: Basic (Difficulty ★★☆☆☆)

A pyramid has a square base of side $6$ and height $5$. Its volume is:

Options:

• A. $180$
• B. $60$
• C. $30$
• D. $90$

Detailed Solution:

1. Base area: $S_{\text{base}} = 6^2 = 36$

2. Apply the pyramid volume formula: $V = \frac{1}{3}(36)(5) = \frac{180}{3} = 60$

Option A is the prism volume — exactly the error of forgetting the $\frac{1}{3}$, and it is placed first deliberately.

Answer: B

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

A regular square pyramid has base edge $6$ and height $4$. Find its lateral surface area.

Options:

• A. $60$
• B. $48$
• C. $96$
• D. $72$

Detailed Solution:

1. The apothem of the square base (centre to the midpoint of a side) is half the side: $r = \frac{6}{2} = 3$

2. Find the slant height using the height–apothem–slant triangle: $l' = \sqrt{h^2 + r^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5$

3. Base perimeter: $c = 4 \times 6 = 24$

4. Lateral surface area: $S_{\text{lateral}} = \frac{1}{2}(24)(5) = 60$

Check by another route: each lateral face is a triangle with base $6$ and height $l' = 5$, giving area $\frac{1}{2}(6)(5) = 15$. Four faces: $4 \times 15 = 60$ ✓

Answer: A

Common trap: using the height $4$ instead of the slant height $5$ in the lateral-area formula gives $\frac{1}{2}(24)(4) = 48$, which is option B.

Common Mistakes

❌ Mistake 1: Forgetting the $\frac{1}{3}$

The single most-penalised error in solid geometry. Prism: $V = Sh$. Pyramid: $V = \frac{1}{3}Sh$.

❌ Mistake 2: Using the height instead of the slant height for lateral area

$S_{\text{lateral}} = \frac{1}{2}c \cdot l'$ needs the slant height $l'$, which is always longer than $h$.

❌ Mistake 3: Confusing slant height with lateral edge

The slant height goes to the midpoint of a base edge; the lateral edge goes to a vertex. They are different lengths, related by $l^2 = l'^2 + \frac{a^2}{4}$.

❌ Mistake 4: Assuming any pyramid is regular

Unless the question says 正棱锥, the apex need not be above the centre — and none of the regular-pyramid shortcuts apply.

Study Tips

1. ✅ Write the $\frac{1}{3}$ first, before anything else, whenever you see 棱锥
2. ✅ Draw the height–apothem–slant right triangle — it solves most regular-pyramid questions
3. ✅ Label $h$, $r$, $R$, $l'$, $l$ on your diagram to avoid substituting the wrong length
4. ✅ Cross-check lateral area by computing one face and multiplying by the number of faces
5. ✅ Remember the prism relationship: a pyramid is one third of the prism on the same base

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💡 Exam Tip: 正棱锥及其有关计算 appears regularly in CSCA solid geometry, and the "prism vs pyramid" volume contrast is one of the most reliable sources of easy marks — or easy losses.