chapter 12 – surface area and volume of solids section 12.1– space figures and nets

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Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

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Page 1: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Chapter 12 – Surface Area and Volume of Solids

Section 12.1– Space Figures and Nets

Page 2: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.1

Polyhedron – a 3-D figure whose surfaces are polygons.

Face – individual polygon of the polyhedron.

Edge – is a segment that is formed by the intersection of two faces.

Vertex – is a point where three or more edges intersect.

Page 3: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.1

Net – a 2-D pattern that you can fold to form a 3-D figure.

Euler’s Formula – the number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula:

F + V = E + 2

Page 4: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CUBE: Net Drawing

Page 5: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CUBE: 3-Dimensional

Faces

Edge

Vertex

Page 6: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CYLINDER: Net Drawing

Page 7: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CYLINDER: 3-Dimensional

Faces

Edge

Page 8: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

TRIANGULAR PRISM: Net Drawing

Page 9: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

TRIANGULAR PRISM: 3-Dimensional

Faces Edge

Vertex

Page 10: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

RECTANGULAR PRISM: Net Drawing

Page 11: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

RECTANGULAR PRISM: 3-Dimensional

Faces

Edge

Vertex

Page 12: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

HEXAGONAL PRISM: Net Drawing

Page 13: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

HEXAGONAL PRISM: 3-Dimensional

Faces

Edge

Vertex

Page 14: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

TRIANGULAR PYRAMID: Net Drawing

Page 15: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

TRIANGULAR PYRAMID: 3-Dimensional

Slant HeightAltitude

Page 16: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

SQUARE PYRAMID: Net Drawing

Slant Height

Page 17: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

SQUARE PYRAMID: 3-Dimensional

Slant Height

Page 18: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

HEXAGONAL PYRAMID: Net Drawing

Page 19: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

HEXAGONAL PYRAMID: 3-Dimensional

Slant Height

Altitude

Page 20: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Chapter 12 – Surface Area and Volume of Solids

Section 12.2 – Surface Areas of Prisms and Cylinders

Page 21: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.2

Prism – is a polyhedron with exactly two congruent, parallel faces.

Bases – two congruent, parallel faces of a prism.

Lateral Faces – additional faces of a prism.

Altitude – is a perpendicular segment that joins the planes of the bases.

Page 22: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.2

Height – the length of the altitude.Right Prism – the lateral faces are

rectangles and a lateral edge is the altitude of the prism.

Oblique Prism – at least one lateral face is not a rectangle.

Lateral Area – is the sum of the area of the lateral faces.

Page 23: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CUBE: 3-Dimensional

BASE

LATERALFACE

Page 24: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

RECTANGULAR PRISM: 3-Dimensional

BASE

LATERALFACE

Page 25: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

TRIANGULAR PRISM: 3-Dimensional

BASE

LATERALFACE

Page 26: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

HEXAGONAL PRISM: 3-Dimensional

BASE

LATERALFACE

Page 27: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

OBLIQUE PRISM: 3-Dimensional

BASE

LATERALFACE

ALTITUDE

Page 28: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.2

Surface Area – the sum of the lateral area and the two bases.

Theorem 10-1 – the lateral area of a right prism is the product of the perimeter of the base and the height.

L.A. = phThe surface area of a right prism is the sum of the lateral area and the area of the 2 bases.

S.A. = L.A. + 2B

Page 29: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.2

Cylinder – is a three-dimensional figure with exactly two congruent, parallel faces.

Bases – two congruent, parallel faces of a cylinder are circles.

Altitude – is a perpendicular segment that joins the planes of the bases.

Page 30: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CYLINDER: 3-Dimensional

BASE

Page 31: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

OBLIQUE CYLINDER: 3-Dimensional

BASE

ALTITUDE

Page 32: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.2

Surface Area – the sum of the lateral area and the two circular bases.

Theorem – the lateral area of a right prism is the product of the circumference of the base and the height of the cylinder.

L.A. = 2πrh or L.A. = πdhThe surface area of a right prism is the

sum of the lateral area and the area of the 2 bases.

S.A. = L.A. + 2B or S.A. = 2πrh + 2πr2

Page 33: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Chapter 12 – Surface Area and Volume of Solids

Section 12.3 – Surface Areas and Pyramids and Cones

Page 34: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Moving from Prisms/Cylinders to Pyramids/Cones

Page 35: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.3

Pyramid – is a polyhedron in which one face can be any polygon and the other faces are triangles that meet at a common vertex.

Bases – the only face of a pyramid that is not a triangle.

Lateral Faces – triangles of pyramid.Vertex of a pyramid – the point where all

lateral faces of a pyramid meet.

Page 36: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.3

Altitude – is a perpendicular segment from the vertex to the plane of the base.

Height – the length of the altitude (h).Regular Pyramid – a pyramid whose base is

a regular polygon and whose lateral faces are congruent isosceles triangles.

Slant Height – is the length of the altitude of a lateral face of a pyramid.

Lateral Area – is the sum of the area of the congruent lateral faces.

Page 37: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

TRIANGULAR PYRAMID: 3-Dimensional

Slant HeightAltitude

Page 38: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

SQUARE PYRAMID: 3-Dimensional

Slant Height

Page 39: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

HEXAGONAL PYRAMID: 3-Dimensional

Slant Height

Altitude

Page 40: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.3

Surface Area – the sum of the lateral area and the area of the base.

Theorem – the lateral area of a regular pyramid is the half the product of the perimeter of the base and the slant height.

L.A. = ½ plThe surface area of a regular pyramid is the sum of the lateral area and the area of the base.

S.A. = L.A. + B

Page 41: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.3

Cone – is a “pointed” like a pyramid, but its base is a circle.

Right Cone – the altitude is a perpendicular segment from the vertex to the center of the base.

Bases – the only circle on a cone.

Vertex of a cone – the only distinctive point on the object.

Page 42: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.3

Altitude – is a perpendicular segment from the vertex to the plane of the base.

Height – the length of the altitude (h).

Slant Height – is the distance from the vertex to a point on the edge of the base.

Lateral Area – is ½ the perimeter (circumference) of the base times the slant height.

Page 43: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CONE: Net Drawing

Page 44: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

CONE: 3-Dimensional

Page 45: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.3

Surface Area – the sum of the lateral area and the area of the base.

Theorem – the lateral area of a right cone is the half the product of the circumference of the base and the slant height.

L.A. = ½ 2rl or rl The surface area of a right cone is the sum of the lateral area and the area of the base.

S.A. = L.A. + B

Page 46: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Chapter 12 – Surface Area and Volume

Section 12.6 – Surface Area and Volumes of Spheres

Page 47: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.6Sphere

Set of all points equidistant from a given point.

C

Page 48: Chapter 12 – Surface Area and Volume of Solids Section 12.1– Space Figures and Nets

Section 12.6Surface Area of a Sphere

S = 4πr 2

C