Points, lines and planes: the basic rules (axioms)
A point has position but no size. A line goes on forever in two directions. A plane is a flat surface that goes on forever in all directions, like a tabletop with no edges.
Solid geometry starts from a few rules we accept without proof, called axioms:
- Through any three points not on one line there is exactly one plane.
- If two points of a line lie in a plane, the whole line lies in that plane.
- If two different planes share a point, they meet in a whole line.
From these we get: a plane is also fixed by a line and a point not on it, by two intersecting lines, or by two parallel lines.
Two lines in space
In a flat plane, two lines either meet or are parallel. In space there is a third choice.
- Intersecting lines meet at exactly one point.
- Parallel lines lie in one plane and never meet.
- Skew lines are not in any one plane. They never meet and are not parallel.
In the cube ABCD-EFGH, AB and CG are skew. The angle between skew lines is found by sliding one of them parallel until they meet. AB slides to DC, and DC ⊥ CG, so the angle between AB and CG is 90°.
Rule: two lines that are each parallel to a third line are parallel to each other.
A line and a plane, two planes
A line and a plane can be in three positions: the line lies in the plane, cuts it at one point, or is parallel to it (no common point).
Test for a line parallel to a plane: if the line is parallel to some line inside the plane, it is parallel to the plane. EF ∥ AB and AB is in ABCD, so EF ∥ ABCD.
Two planes are either parallel or meet in a line. Test: if two intersecting lines of one plane are parallel to the other plane, the planes are parallel.
A section of a solid is the shape you get when a plane cuts it, like slicing a loaf. A plane parallel to a cube face gives a square section; a slanted cut can give a rectangle, triangle or hexagon.
Parallel projection (the way we draw 3D objects on paper) keeps parallel lines parallel and keeps the ratio of lengths on one line.
Perpendicular lines and planes
A line is perpendicular to a plane if it is perpendicular to every line of the plane through its foot.
Test: it is enough to check two intersecting lines in the plane. AE ⊥ AB and AE ⊥ AD, so AE ⊥ plane ABCD.
The orthogonal projection of a point on a plane is the foot of the perpendicular from it. The projection of a line is its "shadow" under light falling straight down.
Theorem of three perpendiculars: a line in the plane is perpendicular to a slanted line if and only if it is perpendicular to that line's projection.
The distance from a point to a plane is the length of the perpendicular. The distance between two parallel planes is the same everywhere.
Angles in space: line–plane and dihedral
The angle between a line and a plane is the angle between the line and its projection on the plane. For the cube diagonal AG, the projection is AC, so θ = ∠GAC. It is between 0° and 90°.
A dihedral angle is the opening between two half-planes that share an edge, like a half-open book. To measure it, draw a line in each face perpendicular to the edge at the same point; the angle between them is the dihedral angle.
Three or more planes meeting at one point form a polyhedral angle (three faces: a trihedral angle, like the corner of a room).
Method: find a right triangle that contains the angle, then use Pythagoras and tan, sin or cos.
Space around us: latitude, longitude and nets
A point in a cuboid can be given by three coordinates (length, width, height) from one corner. G in a cube of side 3 with A at the origin is (3, 3, 3).
On a sphere like the Earth, a point is fixed by two angles: latitude (north or south of the equator) and longitude (east or west of a chosen half-circle). Circles of latitude are sections of the sphere by planes parallel to the equator.
A net is a flat pattern that folds into a solid. A square pyramid's net is a square with four triangles; a cone's net is a circle and a sector of a larger circle.
Try it: a skew-line model
Take a shoebox. Stick a straw along one bottom edge and another straw along an upright edge that does not touch it. Look from every side: the straws never meet, and they are not parallel. They are skew. Now hold a pencil straight up on the box lid and check with a set square that it makes 90° with two lid edges. It is perpendicular to the lid. Compare with steps 2 and 4 of the 3D.
Key formulas and definitions
- 3 non-collinear points → exactly 1 plane
- Line ⊥ plane ⇔ line ⊥ two intersecting lines of the plane
- Cube diagonal: face diagonal = a√2, space diagonal = a√3
- Line–plane angle: tan θ = height / projection
- Cuboid diagonal = √(l² + b² + h²)
Worked examples
1. How many planes pass through three points that lie on one line?
Infinitely many. The three points are all on one line, and a line lies in endless planes (like pages of a book around the spine). Only three points NOT on one line fix a single plane.
2. In cube ABCD-EFGH, say whether each pair is parallel, intersecting or skew: (a) AB, GH (b) AE, BF (c) AB, FG (d) AC, BD.
(a) AB ∥ DC ∥ HG, so parallel. (b) Both upright edges: parallel. (c) AB is on the bottom, FG on the top, going different ways: skew. (d) The two face diagonals of ABCD cross at the centre: intersecting.
3. Find the angle between the skew lines AB and FG in a cube.
Slide FG down parallel to BC. The angle between AB and BC is 90°. So the angle between AB and FG is 90°.
4. A cube has edge 4 cm. Find the length of the space diagonal AG and its angle with the base.
AC = √(4² + 4²) = 4√2 ≈ 5.66 cm. AG = √(AC² + CG²) = √(32 + 16) = √48 = 4√3 ≈ 6.93 cm. tan θ = CG/AC = 4/(4√2) = 1/√2, so θ ≈ 35.3°.
5. A 6 m pole stands straight on flat ground. A wire goes from the top to a point 8 m from the foot. Find the wire length and its angle with the ground.
The pole ⊥ ground, so the triangle is right-angled at the foot. Wire = √(6² + 8²) = 10 m. tan θ = 6/8 = 0.75, θ ≈ 36.9°.
6. In a cuboid with base 3 × 4 and height 5, find the dihedral angle between plane ABGH and the base (AB = 3, BC = 4).
AB is the common edge. BC ⊥ AB in the base and BG ⊥ AB in the slanted plane. So the angle is ∠GBC. tan φ = CG/BC = 5/4, φ ≈ 51.3°.
Common mistakes
- Thinking two lines that do not meet must be parallel. In space they may be skew.
- Checking only one line in the plane to prove a line is perpendicular to the plane. You need two intersecting lines.
- Measuring the line–plane angle against any line in the plane instead of against the projection (shadow).
- Mixing up the face diagonal a√2 and the space diagonal a√3 of a cube.