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.

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.

Types of pyramid1.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.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.

Surface area of PyramidThe 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 LP= perimeter of the baseL= apothem

Total surface area=1/2 (P) x L + BP= perimeter of the baseL= apothemB= area of the base

The perimeter of the base is the sum of the sides.p = 3(8) = 24 cmLateral 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.

Volume of PyramidFormula :1/3 B x HB=base of pyramidH=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

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.

TRUNCATED SQUARE PYRAMID

TRUNCATED PENTAGONAL

PYRAMID

EXAMPLE OF TRUNCATED PYRAMIDS

ApothemHeight of any of its sides.

Smaller Base, A

Bigger Base, B

HeightPerpendicular distance between the base

Lateral facesTrapezoids

Pyramidal Frustum

APOTHEMTo 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

AREA • LATERAL AREA

• TOTAL SURFACE AREA

FORMULA:

P = Perimeter of Bigger BaseP1 = Perimeter of Smaller BaseAp= Apothem of the Frustum

FORMULA:

P = Perimeter of Bigger BaseP1 = Perimeter f Smaller BaseAp= Apothem of the FrustumA = Area of the bigger base areaA1 = Area of smaller base area

VOLUMEFORMULA :

V = 1/3 (a2 + ab + b2 ) h