Introduction

Geometrical properties of common solids
 A geometrical figure having length only is in one dimension
 A figure having area but not volume is in two dimension
 A figure having vertices ( points),edges(lines) and faces (plans) is in three dimension
Examples of Three Dimensional Figures
Rectangular Prism
 A threedimensional figure having 6 faces, 8 vertices, and 1 2 edges
Triangular Prism
 A threedimensional figure having 5 faces, 6 vertices, and 9 edges.
Cone
 A three dimensional figure having one face.
Sphere
 A three dimensional figure with no straight lines or line segments
Cube
 A three dimensional figure that is measured by its length, height, and width.
 It has 6 faces, 8 vertices, and 12 edges
Cylinder
 A three dimensional figure having 2 circular faces
Rectangular Pyramid
 A threedimensional figure having 5 faces, 5 vertices, and 8 edges
Angle Between a Line and a Plane
 The angle between a line and a plane is the angle between the line and its projection on the plane

The angle between the line L and its projection or shadow makes angle A with the plan. Hence the angle between a line and a plane is A.
Example
The angle between a line, r, and a plane, π, is the angle between r and its projection onto π, r'.
Example
Suppose r' is 10 cm find the angle α
Solution
To find the angle we use tanθ = opposite = 4 = 0.4
adjacent 10
tan^{1}(0.4) = 21.8^{o}
Angle Between Two Planes
 Any two planes are either parallel or intersect in a straight line.

The angle between two planes is the angle between two lines, one on each plane, which is perpendicular to the line of intersection at the point
Example
The figure below PQRS is a regular tetrahedron of side 4 cm and M is the mid point of RS; Show that PM is 2√3 cm long, and that triangle PMQ is isosceles
 Calculate the angle between planes PSR and QRS
 Calculate the angle between line PQ and plane QRS
Solution
Triangle PRS is equilateral. Since M,is the midpoint of RS , PM is perpendicular bisector
PM^{2 }= 4^{2 } 2^{2}= 12
PM = √12 cm
= √(4 x 3) = 2√3 cm
Similar triangle MQR is right angled at M
QM^{2}= 4^{2 } 2^{2}= 12
QM = √12 cm
= 4 x 3 = 2√3 cm
Since PM = QM = √12 cm Triangle PMQ is isosceleles 
The required angle is triangle PMQ .Using cosine rule
4^{2 }= (2√3)^{2 }+ (2√3)^{2 } 2(2√3)(2√3)cos m
16 = 12 + 12  2 x 12cos m
= 24  24cos m
cos m= 2416 = 0.3333
24
Therefore, m = 70.53^{o} 
The required angle is triangle PQM
Since triangle PMQ is isosceles with triangle PMQ = 70.54^{o};
∠PQM = ½ (180  70.54)
= ½(109.46)
= 54.73^{o}
Past KCSE Questions on the Topic

The diagram below shows a right pyramid VABCD with V as the vertex. The base of the pyramid is rectangle ABCD, WITH ab = 4 cm and BC= 3 cm. The height of the pyramid is 6 cm.

Calculate the
 Length of the projection of VA on the base
 Angle between the face VAB and the base

P is the mid point of VC and Q is the mid – point of VD. Find the angle between the planes VAB and the plane ABPQ

Calculate the

The figure below represents a square based solid with a path marked on it. Sketch and label the net of the solid.

The diagram below represents a cuboid ABCDEFGH in which FG= 4.5 cm, GH = 8 cm and HC = 6 cm
Calculate: The length of FC

 The size of the angle between the lines FC and FH
 The size of the angle between the lines AB and FH
 The size of the angle between the planes ABHE and the plane FGHE

The base of a right pyramid is a square ABCD of side 2a cm. The slant edges VA, VB, VC and VD are each of length 3a cm.
 Sketch and label the pyramid
 Find the angle between a slanting edge and the base

The triangular prism shown below has the sides AB = DC = EF = 1 2 cm. the ends are equilateral triangles of sides 1 0cm. The point N is the mid point of FC.
Find the length of:
 BN
 EN
 Find the angle between the line EB and the plane CDEF
