- Find volumes and surface areas of prisms, pyramids, cylinders, cones and balls, including cone nets and spheres around boxes.
- Decide whether lines and planes in space are parallel, perpendicular or skew, using the criteria and properties.
- Work in the rectangular coordinate system in space: distances, midpoints and symmetric points.
- Use space vectors to find angles between lines and between a line and a plane, and to prove perpendicularity.
A room is 12 m long, 4 m wide and 3 m high. A fly sits in a corner on the floor and flies straight to the opposite corner of the ceiling. How far does it fly? Not 12 + 4 + 3 = 19 m, and not 12 m either: the answer is √(12² + 4² + 3²) = 13 m. Solid geometry (立体几何) is plane geometry with one more dimension, and the CSCA syllabus lists it in the “Geometry and algebra” part as “rectangular coordinate system in space and properties of simple solids”. In this lesson you learn the volumes and surface areas of simple solids, the rules for parallel and perpendicular lines and planes, coordinates in space, and how space vectors turn angle questions into arithmetic. Plane vectors themselves are in “Plane vectors”, and distances in the plane in “Lines and circles in the coordinate plane”.
Simple solids: volume and surface area
A prism (棱柱) has two congruent polygons in parallel planes as its bases, and its lateral faces are parallelograms; in a right prism the lateral edges are perpendicular to the bases, so the lateral faces are rectangles (a rectangular box and a cube are right prisms). A pyramid (棱锥) has one polygon as its base and triangular lateral faces that meet at the apex. In a regular pyramid the base is a regular polygon and the apex lies directly above its centre; the height of a lateral face is called the slant height (apothem).
These are solids of revolution. A cylinder (圆柱) is formed by turning a rectangle about one of its sides, a cone (圆锥) by turning a right triangle about a leg, and a ball (球) by turning a semicircle about its diameter; the surface of a ball is a sphere. Every segment from the apex of a cone to the circle of its base is a slant height (generatrix, 母线) l, and the height h, the radius r and l form a right triangle: l² = r² + h².
Why does a pyramid have exactly one third of the volume of a prism with the same base and height? A cube can be cut into three identical pyramids whose apex is the same vertex of the cube; their bases are the three faces that do not contain that vertex. Each has base a² and height a, and together they fill the cube, so each has volume a³/3 = ⅓ · a² · a. The same factor ⅓ turns a cylinder into a cone.
- Sarea of the base
- hheight: the perpendicular distance between the bases (prism, cylinder) or from the apex to the base (pyramid, cone)
- rradius of the base
- Rradius of the ball
Prisms and cylinders: base × height; pyramids and cones: one third of that. The height is always the perpendicular distance, never a slanting edge.
1) A right prism has a right-triangle base with legs 3 and 4, and its height is 10. Find its volume.
2) A regular square pyramid has base edge 8 and slant height 5. Find its height and volume.
3) A cone has slant height 13 and base radius 5. Find its volume.
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2) The height, half the base edge and the slant height form a right triangle: h = √(5² − 4²) = 3; V = ⅓ · 8² · 3 = 64.
3) h = √(13² − 5²) = 12, V = ⅓ · π · 5² · 12 = 100π.
- r, hradius of the base and height
- lslant height of the cone, l² = r² + h²
- Rradius of the sphere
Unroll the lateral surface: a cylinder gives a rectangle 2πr × h; a cone gives a sector of radius l with arc length 2πr, whose central angle is 360° · r/l. For prisms and pyramids, simply add the areas of all faces.
1) Find the total surface area of a cylinder with r = 2 and h = 5.
2) Find the lateral and total surface area of a cone with r = 3 and h = 4.
3) The lateral surface of a cone unrolls into a sector with a central angle of 120° and radius 6. Find the base radius and the lateral surface area.
4) A sphere has surface area 36π. Find the volume of the ball.
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2) l = √(3² + 4²) = 5; lateral area π · 3 · 5 = 15π, total 9π + 15π = 24π.
3) The arc of the sector is the circumference of the base: 2π · 6 · (120°/360°) = 4π = 2πr, so r = 2; lateral area πrl = π · 2 · 6 = 12π.
4) 4πR² = 36π ⇒ R = 3; V = 4π · 27/3 = 36π.
A ball is inscribed in a cylinder: it touches both bases and the lateral surface. What is the ratio of the volume of the ball to the volume of the cylinder?
A) 1 : 3 B) 2 : 3 C) 4 : 3 D) 3 : 2
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A is the ratio of a cone to a cylinder with the same base and height. C takes the height of the cylinder as R instead of 2R. D is the inverse ratio (cylinder : ball).
Lines and planes in space
In the plane two different lines either intersect or are parallel. In space there is a third possibility, and most CSCA traps in this topic come from it. The edges of a cube are the best model: in the cube ABCD-A₁B₁C₁D₁ the bottom face is ABCD, the top face is A₁B₁C₁D₁, and AA₁, BB₁, CC₁, DD₁ are the vertical edges.
Skew lines (异面直线) are lines that neither intersect nor are parallel, so no plane contains both of them; in the cube, AB and CC₁ are skew. The angle between skew lines is found by moving one of them parallel to itself until the two meet; it lies in the range 0° < θ ≤ 90°.
A line is perpendicular to a plane (线面垂直) if it is perpendicular to every line of the plane. The angle between a slanting line and a plane is the angle between the line and its projection on the plane (0° ≤ θ ≤ 90°).
In practice nobody checks “every line”. The criteria (判定定理) in the table reduce each relation to one or two lines, and the properties (性质定理) tell you what you may use once the relation is known.
| Relation | Criterion (how to prove it) | Property (what follows) |
|---|---|---|
| Line ∥ plane | a line outside the plane is parallel to a line of the plane | a plane through the line cuts the given plane along a line parallel to it |
| Plane ∥ plane | two intersecting lines of one plane are parallel to the other plane | a third plane cuts them along two parallel lines |
| Line ⊥ plane | the line is perpendicular to two intersecting lines of the plane | it is perpendicular to every line of the plane; two lines perpendicular to the same plane are parallel |
| Plane ⊥ plane | one plane contains a line perpendicular to the other plane | a line in one plane that is perpendicular to their common line is perpendicular to the other plane |
In the cube ABCD-A₁B₁C₁D₁:
1) Name the edges that are skew to AB.
2) Prove that BD ⊥ AC₁.
3) Is the line A₁B parallel to the plane DCC₁D₁?
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2) BD ⊥ AC (diagonals of a square) and BD ⊥ AA₁ (AA₁ is perpendicular to the base). AC and AA₁ intersect, so BD ⊥ plane ACC₁A₁, and therefore BD is perpendicular to every line of that plane, including AC₁.
3) A₁B ∥ D₁C (A₁BCD₁ is a rectangle); D₁C lies in the plane DCC₁D₁ and A₁B does not, so yes, A₁B ∥ plane DCC₁D₁.
m and n are two different lines, and α and β are two different planes. Which statement is correct?
A) If m ∥ α and n ∥ α, then m ∥ n.
B) If α ⊥ β and m ⊂ α, then m ⊥ β.
C) If m ⊥ n and n ⊂ α, then m ⊥ α.
D) If m ⊥ α and n ⊥ α, then m ∥ n.
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A: in the cube, AB and AD are both parallel to the top face, but they intersect. B: only the lines of α that are perpendicular to the common line are perpendicular to β; AB lies in the face ABB₁A₁, which is perpendicular to the base, but AB is not perpendicular to the base. C: one line is not enough; the criterion needs two intersecting lines.
The rectangular coordinate system in space
Three mutually perpendicular number lines with a common origin O form the rectangular coordinate system in space (空间直角坐标系) Oxyz. It is right-handed: if the fingers of the right hand turn from the x-axis to the y-axis, the thumb points along the z-axis. A point P has three coordinates, P(x, y, z). The three coordinate planes are xOy (z = 0), yOz (x = 0) and xOz (y = 0); the projection of P(x, y, z) onto the plane xOy is the point (x, y, 0).
- A(x₁, y₁, z₁), B(x₂, y₂, z₂)two points
- Mmidpoint of AB
- |OP|distance from P(x, y, z) to the origin
The plane formulas from “Lines and circles in the coordinate plane” with a third term. The distance from P(x, y, z) to the z-axis is √(x² + y²), and to the plane xOy it is |z|.
1) Find |AB| and the midpoint of AB for A(−1, 2, 0) and B(3, 0, 4).
2) P(2, −3, 6). Find its distance to the origin, to the z-axis and to the plane xOy.
3) Find the point of the x-axis that is equally far from A(2, 1, 3) and B(4, 3, −1).
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2) |OP| = √(4 + 9 + 36) = 7; to the z-axis √(4 + 9) = √13; to the plane xOy |z| = 6.
3) Let P(x, 0, 0): (x − 2)² + 1 + 9 = (x − 4)² + 9 + 1 ⇒ −4x + 4 = −8x + 16 ⇒ x = 3, so P(3, 0, 0); check: |PA|² = |PB|² = 11.
- a, b, cedges of a rectangular box
- dspace diagonal
- Rradius of the circumscribed sphere (外接球)
- rradius of the ball inscribed in a cube (内切球)
Put the box in coordinates with one vertex at O: the opposite vertex is (a, b, c), so d = |OP|, the distance formula again. The centre of the circumscribed sphere is the midpoint of the diagonal.
1) A rectangular box has edges 2, 3 and 6. Find its space diagonal and the surface area of the sphere through all its vertices.
2) The sphere through the vertices of a cube has surface area 12π. Find the volume of the cube.
3) A ball is inscribed in a cube with edge 2. Find the volume of the ball and the ratio of the volumes of the ball and the cube.
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2) 4πR² = 12π ⇒ R = √3, d = 2√3 = a√3 ⇒ a = 2, V = 8.
3) r = 1, V = 4π/3; the ratio is (4π/3) : 8 = π : 6.
The point P(1, −2, 3) is reflected in the coordinate plane xOz. What are the coordinates of its image?
A) (1, 2, 3) B) (−1, 2, −3) C) (1, 2, −3) D) (−1, −2, 3)
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B is the reflection in the origin, C the reflection in the x-axis, and D the reflection in the plane yOz.
Space vectors: angles and perpendicularity
Proving perpendicularity with the criteria needs a good picture; coordinates need only arithmetic. A vector in space has three coordinates: for A(x₁, y₁, z₁) and B(x₂, y₂, z₂), AB = (x₂ − x₁, y₂ − y₁, z₂ − z₁), end minus start. Addition, scalar multiples and the dot product work exactly as in “Plane vectors”, with one more coordinate. In the cube of the figure A is the origin, and AB, AD and AA₁ lie along the x-, y- and z-axes.
- a = (x₁, y₁, z₁), b = (x₂, y₂, z₂)vectors in space (空间向量)
- ⟨a, b⟩the angle between the vectors, from 0° to 180°
- λa number
For two lines with direction vectors a and b, the angle θ is between 0° and 90°: cos θ = |a · b| / (|a| · |b|). Parallel vectors have proportional coordinates.
1) a = (1, 2, −2), b = (2, −1, 2). Find the cosine of the angle between the vectors and between the lines with these directions.
2) For which k are a = (2, −1, 3) and b = (k, 4, 2) perpendicular? For which m is (−2, 4, m) parallel to (1, −2, 3)?
3) In the cube with edge 1 (as in the figure) find the angle between the skew lines A₁B and B₁C.
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2) 2k − 4 + 6 = 0 ⇒ k = −1. (−2, 4, m) = −2 · (1, −2, 3) ⇒ m = −6.
3) A₁(0, 0, 1), B(1, 0, 0), B₁(1, 0, 1), C(1, 1, 0): A₁B = (1, 0, −1), B₁C = (0, 1, −1). cos θ = 1/(√2 · √2) = 1/2 ⇒ θ = 60°.
- adirection vector of the line
- nnormal vector (法向量) of the plane: a nonzero vector perpendicular to the plane
- θangle between the line and the plane (0° ≤ θ ≤ 90°)
The angle between a and n is 90° − θ, which is why the formula has sin, not cos. The coordinate planes have simple normals: n = (0, 0, 1) for xOy, (1, 0, 0) for yOz and (0, 1, 0) for xOz.
1) In the cube with edge 1 find the angle between the diagonal AC₁ and the base ABCD.
2) Show that AC₁ is perpendicular to the plane A₁BD.
3) The line through A(1, 0, 0) and B(3, 2, 1) makes an angle θ with the plane xOy. Find sin θ.
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2) A₁B = (1, 0, −1), A₁D = (0, 1, −1): AC₁ · A₁B = 1 + 0 − 1 = 0 and AC₁ · A₁D = 0 + 1 − 1 = 0. AC₁ is perpendicular to two intersecting lines of the plane, so AC₁ ⊥ plane A₁BD.
3) AB = (2, 2, 1), |AB| = 3, n = (0, 0, 1): sin θ = 1/3.
- 1Choose the axes
Put the origin at a vertex where three perpendicular edges meet (a corner of a cube or box, or the foot of an edge of a pyramid that is perpendicular to the base).
- 2Write the coordinates
Read them from the edge lengths; for a midpoint, average the coordinates.
- 3Build the vectors
Direction vectors of the lines: end minus start; for a coordinate plane take its normal, for example (0, 0, 1).
- 4Apply the right formula
Two lines: cos θ = |a · b|/(|a||b|); a line and a plane: sin θ = |a · n|/(|a||n|); perpendicularity: dot product 0.
- 5Check the range
An angle between lines, or between a line and a plane, is from 0° to 90°; a negative cosine means you must take the absolute value.
In the rectangular box ABCD-A₁B₁C₁D₁, AB = 2, AD = 1 and AA₁ = 2. What is the cosine of the angle between the lines AD₁ and A₁B?
A) −√10/5 B) √15/5 C) √10/5 D) 2√5/5
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A forgets the absolute value: an angle between lines is never obtuse. B is the sine, not the cosine. D uses |A₁B| = 2 instead of 2√2.
How CSCA asks about this
Solid geometry items are short: one solid given by two or three numbers, a set of statements about lines and planes, or a cube placed in coordinates. Learn the key terms in English and Chinese:
| Term | 中文 | Pinyin |
|---|---|---|
| solid geometry | 立体几何 | lìtǐ jǐhé |
| prism | 棱柱 | léngzhù |
| pyramid | 棱锥 | léngzhuī |
| cylinder | 圆柱 | yuánzhù |
| cone | 圆锥 | yuánzhuī |
| sphere, ball | 球 | qiú |
| cube / rectangular box | 正方体 / 长方体 | zhèngfāngtǐ / chángfāngtǐ |
| surface area | 表面积 | biǎomiànjī |
| volume | 体积 | tǐjī |
| slant height (generatrix) | 母线 | mǔxiàn |
| skew lines | 异面直线 | yìmiàn zhíxiàn |
| line perpendicular to a plane | 线面垂直 | xiànmiàn chuízhí |
| rectangular coordinate system in space | 空间直角坐标系 | kōngjiān zhíjiǎo zuòbiāoxì |
| normal vector | 法向量 | fǎxiàngliàng |
| circumscribed sphere | 外接球 | wàijiēqiú |
Typical question patterns and time-savers (about 75 s per item):
- Volume and area from two numbers: first find the missing length with Pythagoras (l² = r² + h², the height of a regular pyramid from the slant height or a lateral edge), then use the formula; remember ⅓ for pointed solids.
- Cone nets and melting: the arc of the sector is 2πr and its radius is l; melting and recasting keeps the volume.
- Spheres and boxes: 2R = √(a² + b² + c²); for a cube 2R = a√3, and the inscribed ball has r = a/2; lengths × k means volume × k³.
- “Which statement is correct” about lines and planes: test each statement on a cube or on the corner of the room; one counterexample is enough to reject it.
- Coordinates and angles: distance and midpoint formulas, symmetric points (what is named stays); for angle questions put the cube or box in coordinates: |cos| for two lines, sin with a normal vector for a line and a plane, dot product 0 for perpendicularity.
Key points
- V = Sh for prisms and cylinders, V = ⅓Sh for pyramids and cones, V = 4πR³/3 and S = 4πR² for a ball; the lateral area of a cone is πrl with l² = r² + h².
- A cone’s net is a sector of radius l with arc 2πr (central angle 360° · r/l); multiplying lengths by k multiplies areas by k² and volumes by k³.
- In space, lines can be skew; a line is perpendicular to a plane when it is perpendicular to two intersecting lines of the plane.
- |AB| = √(Δx² + Δy² + Δz²); a box’s diagonal √(a² + b² + c²) is the diameter of the sphere through its vertices.
- Angle between lines: cos θ = |a · b|/(|a||b|); line and plane: sin θ = |a · n|/(|a||n|); perpendicular vectors have a · b = 0.
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