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