Three Dimensional Geometry
Geometry · Three Dimensional Geometry
Syllabus tag: KCSE | Mathematics | Form 4 | Topic 3 Three Dimensional Geometry
Lesson objectives
By the end of this topic, you should be able to:
- State the geometric properties of common solids.
- Identify the projection of a line onto a plane.
- Identify skew lines and the angle between them.
- Calculate the angle between a line and a plane, and between two planes.
Three Dimensional Geometry
Angles and lengths inside solids are found by locating the right triangle and applying plane trigonometry to it.
a) Properties of common solids
A prism has a uniform cross-section along its length.
A pyramid has a base and triangular faces meeting at an apex.
In a right pyramid, the apex sits vertically above the centre of the base. That is assumed unless a question says otherwise, and it is what makes the calculations possible.
b) The method
Every three-dimensional problem reduces to a plane triangle.
Sketch the solid, then sketch the relevant triangle separately and larger.
Mark the right angle. Almost every such triangle contains one.
Then use Pythagoras or the trigonometric ratios as usual.
Working on the three-dimensional sketch directly is where errors come from. The separate triangle is worth the extra thirty seconds.
c) Projection of a line onto a plane
The projection of a line onto a plane is its shadow, cast straight down onto the plane.
Drop a perpendicular from each end of the line to the plane. The line joining the feet is the projection.
For a space diagonal of a cuboid, the projection onto the base is the base diagonal.
d) The space diagonal
Apply Pythagoras twice.
First in the base, to get the base diagonal.
Then in the upright triangle formed by the base diagonal, the vertical edge, and the space diagonal.
Equivalently, d² = l² + w² + h² in one step. Both give the same answer, and the two-step version shows the working the mark scheme wants.
e) Angle between a line and a plane
The angle between a line and a plane is measured to its projection on that plane.
So the triangle you need is the line, its projection, and the perpendicular joining them.
That triangle is right-angled at the foot of the perpendicular, so the ratios apply directly.
Measuring to something other than the projection is the commonest error here. Find the projection first, every time.
f) Angle between two planes
The dihedral angle is found along the line where the planes meet.
From a point on that line, draw a line in each plane perpendicular to the line of intersection.
The angle between those two lines is the angle between the planes.
Both lines must be perpendicular to the line of intersection. Otherwise the angle measured is larger than the true dihedral angle.
g) Skew lines
Skew lines are lines that are neither parallel nor intersecting. They exist only in three dimensions.
In a cuboid, a top edge and a non-matching bottom edge are skew.
To find the angle between skew lines, translate one until it meets the other, keeping its direction. Then measure the angle there.
h) Where this is used
Roof and truss design. Surveying slopes and gradients. Aircraft and ship navigation in three dimensions. Any structure where members meet at angles.
Words to know
- Projection -- the foot of the perpendicular from a point to a plane, or the line joining such feet.
- Skew lines -- lines that are neither parallel nor intersecting.
- Dihedral angle -- the angle between two planes.
- Slant height -- the distance from the apex to the midpoint of a base edge.
- Slant edge -- the distance from the apex to a base corner.
:::checkpoint Check yourself
- Find the space diagonal of a cuboid 12 cm by 4 cm by 3 cm.
- What is the projection of a space diagonal onto the base?
- How is the angle between a line and a plane defined?
- What are skew lines? :::
Bridge to practice
The exercises begin with solids and projections, move through angles between lines and planes and between two planes, and finish with lengths and applied problems. For every question, redraw the working triangle as a flat figure with its right angle clearly marked.