Basic DefinitionCavalieri’s Theorem and Volume

Prism

Chapter IIISolid of V = Bh

Ireneo G. Ignacio

December 11, 2008

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Outline

1 Basic Definition

2 Cavalieri’s Theorem and Volume

3 PrismParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Outline

1 Basic Definition

2 Cavalieri’s Theorem and Volume

3 PrismParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Outline

1 Basic Definition

2 Cavalieri’s Theorem and Volume

3 PrismParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Definition

Plane is a collection of straight lines parallel to each other. Twoplanes are parallel if they don’t share the same line. A line isparallel to a given plane provided it doesn’t have a point on theplane however we extend its end points.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Definition

When two planes meet, the intersection is a line called edge. Theangle formed by any two intersecting line is the dihedral angles.And the intersecting planes are the faces.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Definition

When three or more planes meet at a point then they form apolyhedral angle and the point of intersection is called vertex.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Definition

Solid is any limited portion of space, bounded by surfaces. When asolid was cut by a plane the plane figure formed is called sectionof a solid. This section is called right section provided that planepassed through the plane is perpendicular to all its edges. Apolyhedron is a solid bounded by planes.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Theorem

If in two solids of equal altitude the sections made by planesparallel to and at the same distance from their respective bases arealways equal, the volumes of the solids are equal.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

Theorem

If the bases of a solid are equal in area and lie in parallel planesand every section of the solid parallel to the base is equal in areato that of the base, the volume of the solid is the product of itsbase and altitude.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Definition

Prism is a polyhedron of which two faces are equal polygons inparallel planes, and the other faces are parallelograms.

Parallel facesif called the bases. Prisms are named according to its base (e.g.triangular prism has triangle as its base)

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Definition

Prism is a polyhedron of which two faces are equal polygons inparallel planes, and the other faces are parallelograms.Parallel facesif called the bases.

Prisms are named according to its base (e.g.triangular prism has triangle as its base)

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Definition

Prism is a polyhedron of which two faces are equal polygons inparallel planes, and the other faces are parallelograms.Parallel facesif called the bases. Prisms are named according to its base (e.g.triangular prism has triangle as its base)

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Properties and Parts

1 Bases are equal parallel polygons.

2 Altitude is the distance between the bases.

3 Faces which are not the base is called lateral faces.

4 The intersection of two adjacent lateral faces is the lateraledge. Any two lateral edges are equal.

5 Right prism is a prism whose lateral edges are perpendicularto the bases.

6 Regular prism is a right prism whose bases are regularpolygon.

7 Lateral surface area is the sum of the areas of its lateralfaces.

8 Total surface area is the sum of all the areas bounding thesolid.

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Example

Given a regular pentagonal prism, with sides of its base is 6cm andan altitude of 7cm. Find

1 Lateral surface area

2 Total surface area

3 Volume4 Suppose the prism is not a right prism but with regular

pentagon as its base as given above.1 find the length of its lateral edges2 find the area of its right section.3 what is the ratio of the product of the length of its lateral

edges and area of its right section to its Volume?

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Example

The trough has triangular ends which lie in parallel planes. Thetop of the trough is a horizontal rectangle 60cm × 100cm and thedepth of the trough is 50cm.(consider the base of the triangularends is 60cm and all the measurements are made inside it)

1 How many cubic centimeter(cc) of water will it hold?

2 How many liters does it contain when the depth of water is35cm

3 What is the depth and the wetted area of water when thetrough contains 13 liters

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Outline

1 Basic Definition

2 Cavalieri’s Theorem and Volume

3 PrismParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Outline

1 Basic Definition

2 Cavalieri’s Theorem and Volume

3 PrismParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh

Basic DefinitionCavalieri’s Theorem and Volume

Prism

ParallelepipedCylinder

Ireneo G. Ignacio Chapter III Solid of V = Bh