Solid geometry extends plane geometry into three dimensions by introducing a new fundamental object: the plane. This chapter establishes the foundational definitions and theorems governing lines and planes in space — perpendicularity, parallelism, dihedral angles, and polyhedral angles — along with Euclid's numbered definitions for the main solid figures.
Key Takeaways
- Three non-collinear points, or two intersecting lines, uniquely determine a plane.
- A line is perpendicular to a plane if it is perpendicular to every line in the plane through its foot; the perpendicular is the shortest line from a point to a plane.
- Two planes perpendicular to the same line are parallel to each other; the intersection of two planes is always a straight line.
- A dihedral angle is measured by the angle between two lines drawn in the respective faces, each perpendicular to the edge at the same point.
- The sum of face angles of any convex polyhedral angle is less than 360°.
What is a Solid?
Solid geometry deals with objects that require an additional dimension — height (or depth) — beyond the length and breadth used in plane geometry. Common examples are the sphere (ball shape) and the rectangular box.
The most basic elements of plane geometry were lines and angles. The corresponding basic elements for solid geometry are planes and solid angles. This article defines these elements and establishes the propositions governing their behaviour in three-dimensional space.
Classical Euclidean Definitions
- A solid is that which has length, breadth, and depth.
- A face of a solid is a surface.
- A plane is a surface such that a straight line joining any two of its points lies wholly on the surface.
- A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane.
- A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane.
- The inclination of a straight line to a plane is the angle contained by the line and its projection on the plane (the line joining the foot of a perpendicular from the elevated endpoint to the point where the original line meets the plane).
- The inclination of a plane to a plane is the acute angle contained by straight lines drawn at right angles to the intersection, one in each plane, at the same point.
- A plane is similarly inclined to another plane as a third is to a fourth when the angles of inclination are equal.
- Parallel planes are those which do not meet.
- A solid angle is the inclination of more than two lines meeting at a common point not all in the same plane — equivalently, the figure contained by more than two plane angles constructed to one point, not all in the same plane.
Lines and Planes
Determining a Plane
A plane is determined by given lines or points if it contains them and no other plane can. We say we pass a plane through the given elements.
- Postulate: One plane, and only one, can be passed through two given intersecting straight lines.
- Corollary: A straight line and a point not on it determine a plane.
- Corollary: Three points not in a straight line determine a plane.
- Corollary: Two parallel lines determine a plane.
Intersection of Planes
The intersection of two planes is the straight line containing all points common to both.
Proposition I — If two planes cut each other, their intersection is a straight line.
Perpendicular to a Plane
A line is perpendicular to a plane if it is perpendicular to every line in the plane passing through its foot.
Proposition II — If a line is perpendicular to each of two other lines at their point of intersection, it is perpendicular to the plane of those two lines.
Proposition III — All perpendiculars to a given line at a given point lie in the plane perpendicular to the line at that point. - Corollary: Through a given point on a line, one and only one plane can be passed perpendicular to the line. - Corollary: Through a given external point, one and only one plane can be passed perpendicular to a given line.
Proposition IV — Through a given point in a plane, one line and only one can be drawn perpendicular to the plane.
Proposition V — Through a given external point, one line and only one can be drawn perpendicular to a given plane. - Corollary: The perpendicular is the shortest line from a point to a plane.
Proposition VI — Oblique lines from a point to a plane, meeting the plane at equal distances from the foot of the perpendicular, are equal; and of two oblique lines meeting the plane at unequal distances, the more remote is the greater. - Corollary: The locus of a point equidistant from two given points is the plane perpendicular to the segment joining them at its midpoint.
Oblique Lines
A line that meets a plane but is not perpendicular to it is oblique to the plane.
Proposition VII — Two lines perpendicular to the same plane are parallel. - Corollary: If one of two parallel lines is perpendicular to a plane, so is the other. - Corollary: If two lines are each parallel to a third line, they are parallel to each other.
Line and Plane Parallel
A line and a plane are parallel if they cannot meet however far produced.
Proposition VIII — If two lines are parallel, every plane containing one (and only one) of them is parallel to the other. - Corollary: Through either of two lines not in the same plane, one plane and only one can be passed parallel to the other.
Parallel Planes
Two planes that cannot meet however far produced are parallel.
Proposition IX — Two planes perpendicular to the same line are parallel.
Proposition X — The intersections of two parallel planes by a third plane are parallel lines. - Corollary: Parallel lines between parallel planes are equal. - Corollary: Two parallel planes are everywhere equidistant.
Proposition XI — A line perpendicular to one of two parallel planes is perpendicular to the other. - Corollary: Through a given point, one plane and only one can be passed parallel to a given plane.
Proposition XII — If two intersecting lines are each parallel to a plane, the plane of those lines is parallel to that plane.
Proposition XIII — If two angles not in the same plane have their sides respectively parallel and on the same side of the line joining their vertices, the angles are equal and their planes are parallel.
Proposition XIV — If two lines are cut by three parallel planes, their corresponding segments are proportional.
Dihedral Angles
The opening between two intersecting planes is a dihedral angle. Its size depends on the amount of rotation needed to bring one face to the other.
Plane angle of a dihedral angle — the angle formed by two straight lines, one in each face, perpendicular to the edge at the same point. This plane angle has the same magnitude from every point on the edge, and may be taken as the measure of the dihedral angle.
Right dihedral angle — if two adjacent dihedral angles formed by one plane meeting another are equal, each is a right dihedral angle. Planes forming a right dihedral angle are perpendicular to each other.
Proposition XV — Two dihedral angles are equal if their plane angles are equal.
Proposition XVI — Two dihedral angles have the same ratio as their plane angles.
Proposition XVII — If two planes are perpendicular to each other, a line in one perpendicular to their intersection is perpendicular to the other.
Proposition XVIII — If a line is perpendicular to a plane, every plane through this line is perpendicular to the plane.
Proposition XIX — If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane.
Proposition XX — The locus of a point equidistant from the faces of a dihedral angle is the plane bisecting the angle.
Proposition XXI — Through a given line not perpendicular to a given plane, one plane and only one can be passed perpendicular to the plane.
Proposition XXII — The projection of a straight line not perpendicular to a plane, upon that plane, is a straight line. - Corollary: The projection of a line perpendicular to a plane is a point.
Inclination of a line to a plane — the acute angle between the line and its projection on the plane.
Proposition XXIII — The acute angle a line makes with its projection on a plane is the least angle it makes with any line in the plane.
Proposition XXIV — Between two lines not in the same plane there is one common perpendicular, and only one. - Corollary: The common perpendicular is the shortest line segment joining the two lines.
Polyhedral Angles
The opening of three or more planes meeting at a common point is a polyhedral angle. A three-faced polyhedral angle is a trihedral angle; four-faced is a tetrahedral angle, and so on.
Proposition XXV — The sum of any two face angles of a trihedral angle is greater than the third.
Proposition XXVI — The sum of the face angles of any convex polyhedral angle is less than four right angles (360°).
Symmetric polyhedral angles — if the faces of a polyhedral angle are produced through the vertex, a symmetric polyhedral angle is formed. In general, two symmetric polyhedral angles are not superimposable (they are mirror images).
Proposition XXVII — Two trihedral angles are equal or symmetric when the three face angles of one are equal respectively to the three face angles of the other. - Corollary: Equal face angles imply equal dihedral angles.
Definitions of Solid Figures
The following definitions from Euclid's Elements (Book XI) complete the foundational vocabulary of solid geometry.
11. A pyramid is a solid figure contained by planes, constructed from one plane to one point.
12. A prism is a solid figure contained by planes, two of which (opposite faces) are equal, similar, and parallel, while the rest are parallelograms.
13. When a semicircle with fixed diameter is carried round and restored to its starting position, the figure comprehended is a sphere. The fixed line is its axis; the centre of the sphere is the same as that of the semicircle; a diameter is any line through the centre terminated by the surface.
14. A cone is formed when a right triangle is rotated about one of its legs. If the fixed leg equals the other leg, the cone is right-angled; if less, obtuse-angled; if greater, acute-angled. The fixed leg is the axis; the base is the circle described by the rotating leg.
15. A cylinder is formed when a rectangle is rotated about one of its sides. The fixed side is the axis; the bases are the circles described by the two opposite sides.
16. Similar cones and cylinders are those in which the axes and base diameters are proportional.
17. A cube is a solid figure contained by six equal squares.
18. An octahedron is a solid figure contained by eight equal equilateral triangles.
19. A tetrahedron is a solid figure contained by four equal equilateral triangles.
20. A dodecahedron is a solid figure contained by twelve equal equilateral pentagons.
21. An icosahedron is a solid figure contained by twenty equal equilateral triangles.
Conclusion
Lines, planes, and solid angles are the three-dimensional analogues of the points, lines, and angles that opened this guide. The conditions for parallelism and perpendicularity between lines and planes, and the measurement of dihedral and solid angles, form the vocabulary without which no solid geometry is possible. Euclid's Book XI definitions — of the pyramid, prism, sphere, cone, and cylinder — set out the full taxonomy of the simple solids whose properties occupy the remaining chapters.
With these foundations in place, the next chapter explores the first family of three-dimensional solids — Prisms and Pyramids.