Types of PyramidsTriangular 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.

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

• 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

• 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.

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.

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:

Volume of a PyramidA 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 bywhere 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

Volume of a Square-based Pyramid

The volume of a square-based pyramid is given by

ExampleExample

A pyramid has a square base of side 4 cm and a height of 9 cm. Find its volume.Solution:

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:

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:

V = volume of frustumH = height of frustumA = area of lower base A’ = area of upper base

VOLUME OF PYRAMIDAL FRUSTUM

Formula :

FRUSTUMthe 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 + 5²h = √(13² - 5² ) = 12cm

V = 12/3 [ 576 + 196 + √(576 x 196)] =4432 cm ³

Ap = 13 cm

Surface Area Of FrustumCalculate 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