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 Prism Pyramid Cone Cylinder 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 .