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

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.

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

CUBE: Net Drawing

CUBE: 3-Dimensional

Faces

Edge

Vertex

CYLINDER: Net Drawing

CYLINDER: 3-Dimensional

Faces

Edge

TRIANGULAR PRISM: Net Drawing

TRIANGULAR PRISM: 3-Dimensional

Faces Edge

Vertex

RECTANGULAR PRISM: Net Drawing

RECTANGULAR PRISM: 3-Dimensional

Faces

Edge

Vertex

HEXAGONAL PRISM: Net Drawing

HEXAGONAL PRISM: 3-Dimensional

Faces

Edge

Vertex

TRIANGULAR PYRAMID: Net Drawing

TRIANGULAR PYRAMID: 3-Dimensional

Slant HeightAltitude

SQUARE PYRAMID: Net Drawing

Slant Height

SQUARE PYRAMID: 3-Dimensional

Slant Height

HEXAGONAL PYRAMID: Net Drawing

HEXAGONAL PYRAMID: 3-Dimensional

Slant Height

Altitude

Chapter 12 – Surface Area and Volume of Solids

Section 12.2 – Surface Areas of Prisms and Cylinders

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.

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.

CUBE: 3-Dimensional

BASE

LATERALFACE

RECTANGULAR PRISM: 3-Dimensional

BASE

LATERALFACE

TRIANGULAR PRISM: 3-Dimensional

BASE

LATERALFACE

HEXAGONAL PRISM: 3-Dimensional

BASE

LATERALFACE

OBLIQUE PRISM: 3-Dimensional

BASE

LATERALFACE

ALTITUDE

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

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.

CYLINDER: 3-Dimensional

BASE

OBLIQUE CYLINDER: 3-Dimensional

BASE

ALTITUDE

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

Chapter 12 – Surface Area and Volume of Solids

Section 12.3 – Surface Areas and Pyramids and Cones

Moving from Prisms/Cylinders to Pyramids/Cones

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.

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.

TRIANGULAR PYRAMID: 3-Dimensional

Slant HeightAltitude

SQUARE PYRAMID: 3-Dimensional

Slant Height

HEXAGONAL PYRAMID: 3-Dimensional

Slant Height

Altitude

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

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.

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.

CONE: Net Drawing

CONE: 3-Dimensional

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

Chapter 12 – Surface Area and Volume

Section 12.6 – Surface Area and Volumes of Spheres

Section 12.6Sphere

Set of all points equidistant from a given point.

C

Section 12.6Surface Area of a Sphere

S = 4πr 2

C