1. Geometry of three-dimensional solids
Chapter 16
Geometry of three-dimensional solids

2. Axiomatic systems

3. Three-point geometry has these rules:
TOK
Geometry was developed along axiomatic systems. This is where everything has to work within a set of rules.
Three-point geometry has these rules:
There exist exactly three different points.
Each pair of points lies on exactly one line.
Each two distinct lines are on at least one point.
What would the axioms be for four-point geometry?

4. Mathematical fact?

5. The angles at the corners of a triangle add up to 180°. That’s a fact.
TOK
The angles at the corners of a
triangle add up to 180°.
That’s a fact.
Isn’t it?
This triangle is drawn on a sphere.
Do its corners add up to 180°?

6. Right-angled triangles in three-dimensional objects

7. Pyramid To work out the height of the triangle face:
Drop a vertical line down from the vertex (top corner)
to the centre of the base, then join the centre of the base
to the midpoint of an edge of the base.

8. Pyramid To work out the length of a diagonal edge from base to vertex:
Drop a vertical line from the vertex down to the centre of the base, and join the centre of the base to a corner of the base.

9. Cuboid
To work out the length of the diagonal across one rectangular face:
Draw from the ends of the diagonal to either of the untouched corners of the rectangle.

10. Cuboid To work out the length of the
three-dimensional diagonal across the box:
From one end, draw along the edge of the box to the next corner. From the other end, go diagonally across the appropriate face to meet the first line.

11. Cone To work out the slant height of the cone:
Drop a vertical line down from the vertex to the centre of the base, and join the centre of the base to the curved edge of the base.

12. Angle between a line and a plane

13. When a line is on a plane, it merges with the plane.
Imagine this line being rotated up off the plane.
This creates a line in a different plane
and thus needs three dimensions.

14. The line should meet the plane or go through it.
Imagine the shadow of the line on the plane.
Draw a perpendicular line from the top of the line to the plane,
so that it meets the shadow of the line.
Use this triangle to work out the angle between the line and the plane.

15. The line should meet the plane or go through it.
Imagine the shadow of the line on the plane.
Draw a vertical line down from the top of the line to the shadow on the plane to make a right angle.
Use this triangle to work out the angle between the line and the plane.