Solid geometry studies figures in space: how points, lines and planes are positioned relative to each other, and the distances and angles between them. The main difficulty is “seeing” the diagram: a drawing on paper distorts space, and right angles are drawn as acute or obtuse. Yet a 3D problem can almost always be reduced to one or several flat right-angled triangles. This topic gives the rules for that reduction and is the foundation for polyhedra, solids of revolution and the coordinate method.
§1. Axioms and determining a plane
The basic notions — point, line, plane — are not defined; their properties are given by axioms:
Through three points not on one line there passes exactly one plane.
If two points of a line lie in a plane, the whole line lies in that plane.
If two planes share a point, they intersect in a line through that point.
From the axioms come three more ways to fix a plane: through a line and a point off it, through two intersecting lines, or through two parallel lines. That is why a three-legged table never wobbles, while a four-legged one may.
§2. Relative positions
Two lines: intersecting, parallel or skew
A line and a plane: intersecting, parallel, or the line lies in the plane
Two planes: intersect in a line or are parallel
Skew lines do not lie in one plane, so they neither meet nor are parallel. This happens only in space. The angle between skew lines is the angle between one of them and a line drawn parallel to the other through a point of the first. The distance between them is the length of their common perpendicular.
Worked example
How are the edges AB and CC₁ of the cube ABCDA₁B₁C₁D₁ positioned? Find the angle and distance between them (edge a).
AB lies in the base plane, CC₁ meets that plane at C, which is not on AB — the lines are skew.
CC₁ ∥ BB₁ and BB₁ ⊥ AB, so the angle is 90°.
BC is perpendicular to both lines and meets both — it is the common perpendicular.
Distance: BC = a.
§3. Parallelism
If a line is parallel to some line in a plane, it is parallel to the plane
If two intersecting lines of one plane are parallel to another plane, the planes are parallel
Key properties:
Two parallel planes are cut by a third plane along parallel lines — the basis for constructing sections.
Segments of parallel lines between parallel planes are equal.
Two lines parallel to a third are parallel to each other (in space too).
Why two intersecting lines? One parallel line is not enough: intersecting planes also contain lines parallel to their line of intersection.
§4. Perpendicularity
If a line is perpendicular to two intersecting lines of a plane, it is perpendicular to the plane
Three perpendiculars theorem: if the projection of an oblique line is perpendicular to a line in the plane, the oblique line itself is perpendicular to it (and conversely)
Two lines perpendicular to the same plane are parallel.
If a plane contains a line perpendicular to another plane, the planes are perpendicular.
The three perpendiculars theorem is the main tool for the distance from a point to a line: instead of hunting for the perpendicular in the drawing, we find it through the projection.
Worked example
ABCD is a square of side 4, MA is perpendicular to its plane, MA = 3. Find the distance from M to the line CD.
MD is an oblique line and AD its projection.
AD ⊥ CD (square), so by the three perpendiculars theorem MD ⊥ CD.
Hence the required distance is MD.
MD = √(3² + 4²) = 5.
§5. Distances and angles
The distance from a point to a plane is the length of the perpendicular. Oblique l, perpendicular h, projection p: l² = h² + p².
The angle between a line and a plane is the angle between the line and its projection: sin φ = h/l, cos φ = p/l.
A dihedral angle is measured by its plane angle — the angle between perpendiculars to the edge drawn in both faces from one point.
Of two obliques from the same point, the one with the longer projection is longer
Worked example
An oblique line is 10 and the point is 6 from the plane. Find the projection and the angle.
p = √(100 − 36) = 8.
sin φ = h/l = 6/10 = 0.6.
Worked example (official sample)
Segment AB does not cross the plane; AA₁ = 2 and BB₁ = 7 are the distances from its ends to the plane. Find the distance from the point dividing AB in the ratio 3 : 2 from A.
The perpendiculars are parallel, so AA₁B₁B is a trapezium.
The perpendicular from the dividing point is parallel to them; the distance changes linearly.
2 + (7 − 2)·3/5 = 2 + 3 = 5.
Worked example
MA is perpendicular to the plane of the square ABCD of side 6, MA = 6. Find the angle between the planes MCD and ABC.
The edge is CD. AD ⊥ CD, so by the three perpendiculars theorem MD ⊥ CD.
The plane angle is ∠MDA.
tg ∠MDA = MA/AD = 6/6 = 1.
Answer: 45°.
§6. Distances in a cube and a cuboid
The cube is the handiest “laboratory” of solid geometry: any two of its edges are parallel, perpendicular or skew, so every theorem can be tested on it.
In a cube (edge a): face diagonal a√2, space diagonal a√3
In a cuboid (dimensions a, b, c): diagonal d² = a² + b² + c²
The diagonal formula comes from applying Pythagoras twice: first the base diagonal √(a² + b²), then the triangle with that diagonal and the height c as legs. The angle between a cube’s diagonal and the base: the projection of the diagonal is the base diagonal a√2 and the height is a, so tg φ = a/(a√2) = 1/√2 (φ ≈ 35.3°).
Worked example
In a cube with edge 2 find the distance from vertex B₁ to the diagonal AC.
BB₁ ⊥ the base, so the projection of B₁ is B.
The perpendicular from B to AC goes to the centre O of the square: BO = a√2/2 = √2.
By the three perpendiculars theorem B₁O ⊥ AC.
B₁O = √(BB₁² + BO²) = √(4 + 2) = √6.
Check yourself
How many planes pass through three points not on one line?
A point is 12 from a plane and an oblique line is 13. Find the projection.
The ends of a segment of length 10 are 3 and 9 from a plane (on the same side). Distance from the midpoint?
An oblique line is 2√2 and its projection 2. Angle with the plane?
How are two lines perpendicular to the same plane positioned?
A cube has edge 4. The distance between AB and CC₁?
The ends of a segment lie on opposite sides of a plane at distances 3 and 9. Distance from the midpoint to the plane?
What are lines called that do not lie in one plane and do not meet?
Common mistakes
Treating skew lines as parallel or intersecting.
Thinking perpendicularity to one line is enough for perpendicularity to a plane — two intersecting lines are needed.
Measuring the angle between a line and a plane with the perpendicular — it is measured with the projection.
Adding distances instead of subtracting when the ends of a segment are on opposite sides of the plane.
Using lines not perpendicular to the edge instead of the plane angle of a dihedral angle.
Trusting the angle in the drawing — in a 3D picture a right angle does not look right.