Geometry
1. Basic Concepts
The building blocks of geometry, rays and line segments, and the definition of finite and infinite sets.
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Finite or infinite?
2. Position report
Determination of the relative positions of spatial elements.
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The mutual position of the elements in space
3. Angles in Space
Angles between lines, planes, and lines and planes, perpendicularity, and determination of angles in space.
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The angle between two planes
4. Near and Far
Distances between points, lines and planes, units of measurement, and determination of the shortest distance.
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Use of units of measurement
5. Locus in Geometry
Determination of specially positioned points and sets of points in the plane and in space.
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Locus of points in space
6. Vectors
Vectors, their magnitude and direction, opposite vectors, and the sum and difference of vectors.
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Multiplying a vector by a scalar
7. Planar Transformations
Basics of point transformations and congruence transformations: translation, rotation, line reflection and point reflection.
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Planar transformations
8. Translation
Vector definition of translation and its distance-, angle- and orientation-preserving properties.
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Translation
9. Reflection
Point and line reflection, fixed points, invariant lines, and the properties of the transformations.
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Reflection about a line in space
10. Similarity
Similarity transformation, scale factor, reduction and enlargement, and ratios of length, area and volume.
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The similarity transformation
11. The Plane as a Mirror
Reflection about a plane, fixed and invariant shapes, and the geometric properties of a plane mirror.
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Reflection about a plane in space
12. Rotation about a Point
Rotation about a point, direction, angle, fixed point, and identity transformations.
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Rotation about a point
13. Rotation in Space
Rotation about a line in space, rotation axis, fixed points, direction, and congruence properties.
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Rotation about a line in space
14. Pythagorean Theorem
Pythagorean theorem, relations between sides of right triangles, proof, and Pythagorean triples.
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Illustration of the Pythagorean theorem
15. Thales' Theorem
Thales’ theorem, its proof, inscribed angle subtended by a diameter, and applications of the theorem.
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Application of Thales’ theorem
16. Theorems abuot (Tri)Angles
Altitude theorem, leg theorem, intercept theorem, related proportions, and the concept of the geometric mean.
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The theorem of parallel intercepting line segments
17. Trigonometry
Trigonometric relations in right triangles: sine, cosine, tangent and cotangent functions.
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Representation of the sine and cosine trigonometric functions
18. Laws of Sines and Cosines
Application of the laws of sines and cosines to determine sides and angles of any triangles.
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Treasure hunt exercise
19. Quadrilaterals
Classification and properties of quadrilaterals: trapezium, parallelogram, rhombus, rectangle, deltoid and square.
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Perimeter and area of quadrilaterals
20. Outside & Inside
Properties of cyclic and tangential quadrilaterals and the related geometric theorems.
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Constructing the circumcircle of a quadrilateral
21. Polygons
Characteristics of polygons, diagonals, interior and exterior angles, and properties of regular polygons.
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The area of a regular polygon
22. Polyhedra
Polyhedra, Euler’s polyhedral formula, and the characteristics of regular solids.
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Solving a math problem: Octahedron
23. Prisms
Definition and parts of prisms, and calculation of their surface area and volume.
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Derivation of a prism
24. Right & Oblique
Comparison of right and oblique prisms, and differences in determining their surface area, height and volume.
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The surface area of a right prism
25. On Quadrilateral Bases
Calculation of surface area and volume of special right prisms with quadrilateral bases, with a practical example.
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The square prism
26. From Column to Cube
Properties, surface area and volume of square prisms and cubes, with a practical Rubik’s Cube example.
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The volume of a cube
27. Pyramid
Definition and parts of pyramids, their derivation, right and oblique pyramids, surface area and volume formulas.
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The volume of a pyramid
28. Truncated Pyramid
Definition and parts of truncated pyramids, surface area and volume, and application of similarity rules.
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The surface area of a truncated pyramid
29. Circles
Basic circle concepts: circumference, radius, diameter, tangent, secant, chord, and calculation of circumference and area.
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The circumference and area of the circle
30. Inside the Circle
Parts of a circle: disc, annulus, sector, segment and semicircle, and determination of their areas.
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Relationships in the circle
31. Angles in the Circle
Central and inscribed angles, their corresponding arcs, and the relations between them.
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Measuring angles
32. Spheres
Characteristics of spheres and spherical surfaces, radius, diameter, and determination of surface area and volume.
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The surface area of a sphere
33. Is the Earth a Sphere?
Geometric relations between the sphere and Earth, geographical coordinates, latitude and longitude.
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The volume of a sphere
34. (Circular) Cylinders
Definition of cylinders, types and parts of circular cylinders, and formulas for surface area and volume.
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The circular cylinder
35. Cones
Parts of cones and cones of revolution: generator, lateral surface, base, and determination of surface area and volume.
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Volume of a circular cone
36. Cones of Revolution
Geometric properties of cones of revolution generated by rotating axially symmetric plane figures.
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The surface area and volume of nested cones
37. Cross Sections of a Cone
Plane sections of cones of revolution: circle, ellipse, parabola and hyperbola, plus truncated-cone derivation and surface area.
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The volume of a truncated cone
38. "Spun" around
Solids of revolution generated by rotating plane figures around a symmetry axis, and examination of their volumes.
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How heavy is the ball?
39. Inscribed Solids
Archimedes’ theorem, and the concept and characteristics of dual solids.
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Spheres inscribed in a cuboid
40. A Solid in a Solid
Joined geometric solids, Platonic solids, and the stellated octahedron.
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A sphere inscribed in a cone
| 40 topics | |
| 102 slides | |
| 1 selftest | |
| 39 test | |
| 163 images | |
| 40 video 3D | |
| View all slides |
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