Unit 4: Surface Area and Volume of Solids Lesson 4.1 Explore Solids Lesson 12.1 from textbook
Objectives: • Determine which of the different three-dimensional figures or solids are polyhedra • Identify parts of the polyhedron such as faces, vertices, and edges. • Identify the solids formed by taking cross sections of a polyhedron.
face Face ______
edge Edge ______
Vertex ______
vertex
Regular, Convex Polyhedron Nonregular, Concave Polyhedron
TYPES OF SOLIDS
Polyhedra Not Polyhedra
CLASSIFYING SOLIDS
To classify a polyhedron, ______.
Polyhedron
Type of Polyhedron
Shape of Base
Name of Polyhedron
Example 1
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find its number of faces, vertices, and edges.
______
Euler’s Theorem
The number of faces, vertices, and edges of a polyhedron are related by the formula
______.
CROSS SECTIONS
Cross Section ______
Example 2
Describe the shape formed by the intersection of the plane and the cube.
______Unit 4.2: Surface Area and Volume of Solids Lesson 4.2 Surface Area of Prisms and Cylinders Lesson 12.2 from textbook
Objectives: • Derive the formula for surface area of a prism and a cylinder using the area formulas for parallelograms and circles. • Use the formulas to calculate the surface area of a prism and a cylinder. • Use the surface area of a prism or cylinder to find the unknown dimensions.
PRISMS
Name of Prism ______
Number of Bases ______
Number of Lateral Faces ______
NET OF A RECTANGULAR PRISM
Lateral Area = ______Base Area = ______Surface Area = ______
OBLIQUE AND RIGHT PRISMS
Surface Area of a Right Prism Theorem Example 1
P = perimeter of base, B = base Area, h = height of prism Find the surface area of the right prism.
S = ______
S = ______
CYLINDERS NET OF A CYLINDER
Shape of Base ______Base Area = ______Lateral Area = ______
Surface Area = ______
Surface Area of a Cylinder Theorem Example 2
r = radius of base, h = height of the cylinder Find the surface area of the cylinder.
S = ______S = ______
Example 3
Find the height of the right cylinder which has a surface area of 157.08 square meters.
h = ______
Unit 4: Surface Area and Volume of Solids Lesson 4.3 Surface Area of Pyramids and Cones Lesson 12.3 from textbook
Objectives: • Derive the formula for surface area of a pyramid and a cone using the area formulas for parallelograms, triangles and circles. • Use the formulas to calculate the surface area of a pyramid and a cone. • Use the surface area of a pyramid or cone to find the unknown dimensions.
PYRAMIDS
NET OF A PYRAMID
l = ______
b = ______
Lateral Area ______
Surface Area = ______
Surface Area of a Pyramid Theorem Example 1
B = base area, P = perimeter of the base, l = slant height Find the lateral area of the pyramid.
S = ______L = ______
Example 2
Find the surface area of the pyramid. S = ______
CONES NET OF A RIGHT CONE
Base Area ______
Lateral Area ______
Surface Area ______
Surface Area for Cones Theorem r = radius of the base, l = slant height of cone
S = ______
Example 3
Find the surface area of the cone.
S = ______S = ______Unit 4: Surface Area and Volume of Solids Lesson 4.4: Volume of Prisms and Cylinders Lesson 12.4 from textbook
Objectives: • Use the formulas to calculate the volume of a prism and a cylinder. • Use the volume of a prism or cylinder to find the unknown dimensions. • Solve problems involving unit conversion for situations involving volumes within the same measurement system.
Volume of a Cube Postulate Volume Congruence Postulate
The volume of a cube is the cube of If two polyhedra are congruent, then the length of its side. their volumes ______.
Volume Addition Postulate.
V = ______The volume of a solid is the sum of the volume of its non-overlapping parts.
Volume of a Prism Volume of a Cylinder
B = Base area, h = height of prism B = Base area, h = height of prism
V = ______V = Bh = ______
*The shape of the base could be a square, rectangle, trapezoid, triangle, or any polygon.
Example 1
Find the volume of the solid.
V = ______V = ______
Example 2
If the volume of the cube is 90 in 3, find the length of each side x.
x = ______x
x x
Cavalieri’s Principle Example 3
Find the volume of the solid.
All three solids have equal heights and cross-sectional areas B. Bonaventura Cavalieri stated that:
______
Example 4 Example 5
Find the volume of the solid. The sculpture is made of 13 beams. The dimensions of each beam are 30 by 30 by 90. Find its volume.
V = ______
V = ______Unit 4: Surface Area and Volume of Solids Lesson 4.5 Volume of Pyramids and Cones Lesson 4.5 from textbook
Objectives: • Use the formulas to calculate the volume of a pyramid and cone. • Use the volume of a pyramid and cone to find the unknown dimensions. • Solve problems involving unit conversion for situations involving volumes within the same measurement system.
Volume of a Pyramid Theorem Volume of a Cone Theorem
V = ______V = ______
Example 1
Find the volume of the solid.
V = ______V ≈ ______
Example 2
If a square base pyramid has a height of 144 meters and a volume of 2,226,450 m 3, what are the dimensions of the base? x = ______
Example 3
Find the volume of the right cone. V ≈ ______
Example 4
Find the volume of the solid shown.
V = ______
Example 5
Unit 4: Surface Area and Volume of Solids Lesson 4.6 Surface Area and Volume of Spheres Lesson 12.6 from textbook
Objectives: • Identify the different parts of the sphere: center, radius, diameter, and great circle. • Use the given formulas to find the surface area and volume of a sphere. • Use the given volume and surface area to find the unknown measures of the sphere. • Solve problems involving unit conversion for situations involving volumes within the same measurement system.
SPHERE GREAT CIRCLES and HEMISPHERES
Surface Area of a Sphere Theorem
S = ______
Example 1 Example 2
Find the surface area of the sphere. Find the radius of the sphere if the surface area is 228 cm 2.
r = ______
S = ______
Example 3
Find the surface area of the composite solid.
S = ______
Volume of a Sphere Theorem Example 4
Find the volume of the sphere.
V = ______
V = ______
Example 5
Find the volume of the composite solid.
V = ______
Example 6