球qiú

Core Concept

A sphere (球) is the set of all points in space at a fixed distance $R$ from a centre $O$. It is the three-dimensional counterpart of the circle, and its equation is the 3-D distance formula squared:

$(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2$

The Two Core Formulas

$V = \frac{4}{3}\pi R^3 \qquad S = 4\pi R^2$

• $V$ — volume (体积)
• $S$ — surface area (表面积)

Keeping them apart: volume has the cube (three dimensions), surface area has the square (two dimensions). The coefficient $\frac{4}{3}$ belongs only to the volume; the surface area coefficient is a clean $4$.

A useful relationship: $\dfrac{dV}{dR} = 4\pi R^2 = S$. The rate at which volume grows equals the surface area — which makes intuitive sense, since expanding the radius adds a thin shell of exactly that area.

Cross-Sections

Any plane cutting a sphere produces a circle. If the plane is at distance $d$ from the centre ($d < R$), the cross-sectional circle has radius:

$r = \sqrt{R^2 - d^2}$

This is the Pythagorean theorem again, and it is the workhorse of CSCA sphere questions. When $d = 0$ the plane passes through the centre and gives the great circle (大圆) of radius $R$ — the largest possible cross-section.

Inscribed and Circumscribed Solids

The most-tested CSCA sphere problems involve a sphere paired with a polyhedron (多面体与球体内切外接问题).

Sphere Circumscribing a Rectangular Box

If a box has edge lengths $a$, $b$, $c$, its circumscribed sphere has the space diagonal as diameter:

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

This single fact solves a large share of circumscribed-sphere questions, because a cube is the special case $a = b = c$:

$2R = a\sqrt{3} \implies R = \frac{\sqrt{3}}{2}a$

Sphere Inscribed in a Cube

The inscribed sphere touches all six faces, so its diameter equals the edge length:

$2R = a \implies R = \frac{a}{2}$

Do not confuse these two. For the same cube of edge $a$, the circumscribed sphere has $R = \frac{\sqrt{3}}{2}a \approx 0.87a$ while the inscribed sphere has $R = 0.5a$.

CSCA Practice Problems

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

Example 1: Basic (Difficulty ★★☆☆☆)

A sphere has surface area $36\pi$. Its volume is:

Options:

• A. $36\pi$
• B. $27\pi$
• C. $12\pi$
• D. $9\pi$

Detailed Solution:

1. From the surface area, recover the radius: $4\pi R^2 = 36\pi \implies R^2 = 9 \implies R = 3$

2. Substitute into the volume formula: $V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (27) = 36\pi$

The numerical coincidence $V = S = 36\pi$ happens only at $R = 3$ and is a favourite exam setup — do not assume it holds in general.

Answer: A

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

A plane cuts a sphere of radius $5$, producing a circular cross-section of area $16\pi$. Find the distance from the centre of the sphere to the plane.

Options:

• A. $2$
• B. $3$
• C. $4$
• D. $\sqrt{21}$

Detailed Solution:

1. From the cross-sectional area, find its radius: $\pi r^2 = 16\pi \implies r = 4$

2. Apply the cross-section relationship $r = \sqrt{R^2 - d^2}$: $4 = \sqrt{25 - d^2}$

3. Square both sides and solve: $16 = 25 - d^2 \implies d^2 = 9 \implies d = 3$

4. Check: $d = 3 < R = 5$, so the plane genuinely cuts the sphere ✓

Recognising the $3$-$4$-$5$ triple ($d$, $r$, $R$) makes this instant.

Answer: B

Common Mistakes

❌ Mistake 1: Swapping the exponents

$V = \frac{4}{3}\pi R^3$ and $S = 4\pi R^2$. Writing $\frac{4}{3}\pi R^2$ mixes both formulas and is the most frequent slip on this topic.

❌ Mistake 2: Confusing inscribed with circumscribed

"内切" means the sphere is inside, touching the faces; "外接" means the sphere is outside, passing through the vertices. Read the Chinese carefully — the two give very different radii.

❌ Mistake 3: Forgetting that a cross-section is a circle, not an ellipse

Every plane section of a sphere is a circle, regardless of the cutting angle.

❌ Mistake 4: Using the diameter where the radius belongs

If a question gives a diameter of $10$, then $R = 5$. Substituting $10$ into $V = \frac{4}{3}\pi R^3$ inflates the answer eightfold.

Study Tips

1. ✅ Memorise both formulas together and check the exponent every time
2. ✅ Draw the right triangle ($d$, $r$, $R$) for cross-section problems
3. ✅ For a cube of edge $a$: circumscribed $R = \frac{\sqrt{3}}{2}a$, inscribed $R = \frac{a}{2}$
4. ✅ Work backwards from $S$ or $V$ to $R$ first — nearly every question needs $R$ as an intermediate step
5. ✅ Re-read 内切 vs 外接 before computing

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💡 Exam Tip: 球的表面积的有关计算 and 球的体积的有关计算 are the two most common CSCA solid-geometry knowledge points involving spheres. Most questions are two steps: recover $R$ from what you are given, then apply the other formula.