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1. Introduction to Geometry
• Geometry is a type of math used to measure
things that are impossible to measure with the
devices.
Geometry can be divided into:
-Plane Geometry is
about flat shapes like
lines, circles and
triangles ... shapes that
can be drawn on a
piece of paper
-Solid Geometry is about
three dimensional objects
like cubes, prisms,
cylinders and spheres.

2. PYRAMID
• The height of the pyramid is the measure of the
perpendicular line segment from the apex to the
base
• The apothem of a regular pyramid is the height of
one of its lateral faces.

3. Types of pyramid
1.Regular pyramid
- its faces are equally sized
2.Irregular pyramid
– faces are not equally sized
3.Convex pyramid
– has convex polygon as its base

4. 4.Concave pyramid
– has concave polygon as its base
5.Right pyramid
– has isosceles triangles as its faces &
apex lies directly above the midpoint
of the base
6. Oblique pyramid
– does not have isosceles triangles as
its lateral sides.

5. Surface area of Pyramid
The total surface area of a regular pyramid is the sum of the areas of its lateral
faces and its base.
Formulas :
Lateral surface area = ½( P ) x L
P= perimeter of the base
L= apothem
Total surface area=1/2 (P) x L + B
P= perimeter of the base
L= apothem
B= area of the base

6. The perimeter of the base is the
sum of the sides.
p = 3(8) = 24 cm
Lateral surface area
=1/2 x 24 x 18
= 216 cm2
***There is no formula for a surface area of a non-
regular pyramid since slant height is not defined. To
find the area, find the area of each face and the area
of the base and add them.
Example :Find the lateral surface area of a regular
pyramid with a triangular base if each edge of the base
measures 8 cm and the slant height( apothem) is 18
cm.

7. Volume of Pyramid
Formula :1/3 B x H
B=base of pyramid
H=height of pyramid
*The height must be
measured as
the vertical distance from the
apex down to the base.
Example :
Solution : 1/3 B x H
1/3(10)(10)
= 233.3cm3

8. PYRAMIDAL
FRUSTUM
PYRAMIDAL
FRUSTUM, also known
as a truncated
pyramid is the result
of a pyramid cut by a
plane parallel to the
base and separating
the apex.

9. TRUNCATED
SQUARE PYRAMID
TRUNCATED
PENTAGONAL
PYRAMID
EXAMPLE OF TRUNCATED
PYRAMIDS

10. Apothem
Height of
any of its
sides.
Smaller
Base, A
Bigger
Base, B
Height
Perpendicular
distance
between the
base
Lateral
faces
Trapezoids
Pyramidal
Frustum

11. APOTHEM
To find the apothem of the pyramidal frustum,
you have to know:-
- The height
- The apothem of the smaller base, Ap1
- The apothem of the bigger base, Ap2
Formula:
The
Height
Bigger Base- Smaller
Base

12. AREA
• LATERAL AREA
• TOTAL SURFACE AREA
FORMULA:
P = Perimeter of Bigger
Base
P1 = Perimeter of Smaller
Base
Ap= Apothem of the
Frustum
FORMULA:
P = Perimeter of Bigger
Base
P1 = Perimeter f Smaller
Base
Ap= Apothem of the
Frustum
A = Area of the bigger
base area
A1 = Area of smaller base
area

13. VOLUME
FORMULA :
V = 1/3 (a2 + ab + b2 ) h