pyramid and frustrum

1. Types of Pyramids
Triangular Pyramid : has a triangle as its base
Square Pyramid : has a square as its base
Pentagonal Pyramid : has a pentagon as its base
Introduction of Pyramid
Pyramid is a solid object where the base is a polygon (a straight-sided flat
shape) and the sides are triangles which meet at the top (the apex). It is a
polyhedron.

2. Right Pyramid : the apex of the pyramid is directly
above the center of its base
Right Pyramid : the apex of the pyramid is directly above
the center of its base
Regular Pyramid : the base of this pyramid is a regular
polygon
Irregular Pyramid : this type of pyramid has an irregular
polygon as its base

3. • Definition Of Surface Area
• The surface area of a three-dimensional figure is
the sum of the areas of all its faces.
• The Surface Area of a Pyramid
• When all side faces are the same:
• [Base Area] + 1/2 × Perimeter × [Slant Length]
• When side faces are different:
• [Base Area] + [Lateral Area]
• Notes On Surface Area
• The Surface Area has two parts: the area of the
base (the Base Area), and the area of the side faces
(the Lateral Area).
• For Base Area :
• It depends on the shape, there are different
formulas for triangle, square, etc.
• For Lateral Area :
• When all the side faces are the same:
• Multiply the perimeter by the "slant length" and
divide by 2. This is because the side faces are
always triangles and the triangle formula is "base
times height divided by 2"
• But when the side faces are different (such as an
"irregular" pyramid) we must add up the area of
each triangle to find the total lateral area.
Surface Area

4. • A pyramidal frustum is a frustum made by
chopping the top off a pyramid.
• The lateral faces of a pyramidal frustum are
trapezoids.
• The height of the pyramidal frustum is the
perpendicular distance between the bases.
• The apothem is the height of any of its sides.

5. Total Surface Area of a Pyramid
A pyramid is a three-dimensional figure
made up of a base and triangular
faces that meet at the vertex, V, which is
also called the apex of the pyramid.
The lower face ABCD is called the base and the perpendicular
distance from the vertex, V, to the base at O is called the height of
the pyramid. The total surface area of a pyramid is the sum of the
areas of its faces including its base.
The number of triangular faces depends on the number of sides of
the base. For example, a pyramid with a rectangular base has four
triangular faces, whereas a pyramid with a hexagonal face is made up
of six triangular faces, and so on.
Note:
•A square pyramid has four equal triangular faces and a square
base.
•A pyramid does not have uniform (or congruent) cross-sections.

6. By Pythagoras' Theorem from right-triangle VOM, we
have
Example
Find the total surface area of a square pyramid with a
perpendicular height of 16 cm and base edge of 24 cm.
Solution:

8. Volume of a Pyramid
A pyramid has a base and triangular sides which rise to
meet at the same point. The base may be any polygon
such as a square, rectangle, triangle, etc.
Volume of a Pyramid
The volume, V, of a pyramid in cubic units is given by
where A is the area of the base and h is the height of the
pyramid.
V=1/3 (area of base)(height)
=1/3 Ah

9. Volume of a Square-based Pyramid
The volume of a square-based pyramid is given by
Example
Example
A pyramid has a square base of side 4 cm and a
height of 9 cm. Find its volume.
Solution:

10. Volume of a Rectangular-based Pyramid
The volume of a rectangular-based pyramid is
given by
Example
Example
Find the volume of a rectangular-based
pyramid whose base is 8 cm by 6 cm and
height is 5 cm.
Solution:

11. Volume of a Triangular Pyramid
The volume of a triangular
pyramid is given by
Example
Find the volume of the following
triangular pyramid, rounding your
answer to two decimal places.
Solution:

12. V = volume of frustum
H = height of frustum
A = area of lower base
A’ = area of upper base
VOLUME OF PYRAMIDAL FRUSTUM
Formula :
FRUSTUM
the portion of a cone or pyramid which remains after its
upper part has been cut off by a plane parallel to its
base, or which is intercepted between two such planes.
13² = h² + 5²
h = √(13² - 5² ) = 12cm
V = 12/3 [ 576 + 196 + √(576 x 196)] =4432 cm ³
Ap = 13 cm

13. Surface Area Of Frustum
Calculate the lateral area, surface area and volume of
the truncated square pyramid whose larger base edge
is 24, smaller base edge is 14 cm and whose lateral
edge is 13 cm.
13² = h² + 5²
h = √(13² - 5² ) = 12cm
Ap = 13 cm