Splash Screen

Polyhedron – a solid with all flat surfaces that enclose a single region Face – the flat surfaces of a polyhedron, they are polygons Edges – the line segments where faces intersect Vertex – a point where three or more edges meet

Concept

Identify Solids A. Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices. Example 1

Identify Solids The solid is formed by polygonal faces, so it is a polyhedron. The two bases are parallel and congruent, therefore it is a prism. The bases are rectangles. This solid is a rectangular prism. Answer: rectangular prism; Bases: rectangles EFHG, ABDC Faces: rectangles FBDH, EACG, GCDH, EFBA, EFHG, ABDC Vertices: A, B, C, D, E, F, G, H Example 1

Identify Solids B. Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices. Example 1

Answer: hexagonal prism; Bases: hexagon EFGHIJ and hexagon KLMNOP Identify Solids The solid is formed by polygonal faces, so it is a polyhedron. The two bases are parallel and congruent, therefore it is a prism. The bases are hexagons. This solid is a hexagonal prism. Answer: hexagonal prism; Bases: hexagon EFGHIJ and hexagon KLMNOP Faces: rectangles EFLK, FGML, GHNM, HNOI, IOPJ, JPKE; hexagons EFGHIJ and KLMNOP Vertices: E, F, G, H, I, J, K, L, M, N, O, P Example 1

Identify Solids C. Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices. Example 1

Answer: Base: circle T Vertex: W no faces or edges Identify Solids The solid has a curved surface, so it is not a polyhedron. The base is a circle and there is one vertex. So, it is a cone. Answer: Base: circle T Vertex: W no faces or edges Example 1

A. Identify the solid. A. triangular pyramid B. pentagonal prism C. rectangular prism D. square pyramid Example 1

B. Identify the solid. A. cone B. cylinder C. pyramid D. polyhedron Example 1

C. Identify the solid. A. triangular prism B. triangular pyramid C. rectangular pyramid D. cone Example 1

Platonic Solids – all of its face are regular congruent polygons and all the edges are congruent Concept

The polygons created by the lateral edges are the Lateral Faces The edges of the prism where the lateral faces intersect are called its lateral edges.  The lateral edges in a prism are congruent and parallel.  Lateral edges: There are 5 congruent and parallel lateral edges in this prism. The polygons created by the lateral edges are the Lateral Faces

Prisms Volume & Surface Area

For volume the units are cubed or its written as cubic units The volume of a prism is the product of the base area times the height of the prism. V = Bh (B = base area,  h = height) For volume the units are cubed or its written as cubic units h = height(altitude) between bases B = area of the base

Find the volume of the triangular prism. Example 2

A net is a two-dimensional figure  that can be cut out and folded up to  make a three-dimensional solid.

The surface area of a prism is the sum of the areas of the bases plus the areas of the lateral faces.  This simply means the sum of the areas of all faces.  Draw each face with the dimensions and find the areas. Add them all together.

Cylinder h = height (altitude) r = radius  

Cylinder h = height (altitude) r = radius  

Cylinder h = height (altitude) r = radius  

Find the Volume, Lateral Area and Surface Area

Pyramids Volume & Surface Area

(B = base area, h = height) The volume of a pyramid is one-third the product of the base area times the height of the pyramid.   (B = base area,  h = height) h = height (altitude) from vertex to base B = area of base

A net is a two-dimensional figure that can be cut out and folded up to make a three-dimensional solid. The surface area of a pyramid is the sum of the area of the base plus the areas of the lateral faces. This simply means the sum of the areas of all faces.  

Find the Volume and Surface Area

Cone h = height (altitude) r = radius s = slant height   The volume of a cone can be calculated in the same manner as the volume of a pyramid:  the volume is one-third the product of the base area times the height of the cone, Since the base of a cone is a circle, the formula for the area of a circle can be substituted into the volume formula for B : (Volume of a cone:  r = radius, h = height) In a right circular cone, the slant height, s, can be found using the Pythagorean Theorem.

Lateral = any face or surface that is not a base. A net is a two-dimensional figure that can be cut out and folded up to make a three-dimensional solid. Lateral = any face or surface that is not a base. In a right circular cone, the slant height, s, can be found using the Pythagorean Theorem: The surface area (of a closed cone) is a combination of the lateral area and the area of the base.  When cut along the slant side and laid flat, the surface of a cone becomes one circular base and the sector of a circle (lateral surface), as shown in the net at the left. Note that the length of the arc in the sector is the same as the circumference of the small circular base. The lateral area (sector) = The base area = area of a circle     Total Surface Area of a Closed Cone = lateral area + base area When working with surface areas of cones, read the questions carefully.    Will the surface area include the base? Will the surface area not include the base?

Find the surface area and volume of the cone. Find Surface Area and Volume Find the surface area and volume of the cone. Example 2

Volume of a cone r = 3, h = 4 Simplify. Use a calculator. Find Surface Area and Volume Volume of a cone r = 3, h = 4 Simplify. Use a calculator. Answer: The cone has a surface area of about 75.4 cm2 and a volume of about 37.7 cm3. Example 2

The volume of a sphere is four-thirds times pi times the radius cubed. (Volume of a sphere:  r = radius) The surface area of a sphere is four times the area of the largest cross-sectional circle (called the great circle).

Surface Area and Volume A. CONTAINERS Mike is creating a mailing tube which can be used to mail posters and architectural plans. The diameter of the base is inches, and the height is feet. Find the amount of cardboard Mike needs to make the tube. The amount of material used to make the tube would be equivalent to the surface area of the cylinder. Example 3

Surface area of a cylinder Surface Area and Volume Surface area of a cylinder r = 1.875 in., h = 32 in. Use a calculator. 399.1 Answer: Mike needs about 399.1 square inches of cardboard to make the tube. Example 3

Surface Area and Volume B. CONTAINERS Mike is creating a mailing tube which can be used to mail posters and architectural plans. The diameter of the base is inches, and the height is feet. Find the volume of the tube. Volume of a cylinder r = 1.875 in., h = 32 in. Use a calculator. 353.4 Example 3

Answer: The volume of the tube is about 353.4 cubic inches. Surface Area and Volume Answer: The volume of the tube is about 353.4 cubic inches. Example 3

A. Jenny has some boxes for shipping merchandise A. Jenny has some boxes for shipping merchandise. Each box is in the shape of a rectangular prism with a length of 18 inches, a width of 14 inches, and a height of 10 inches. Find the surface area of the box. A. surface area = 2520 in2 B. surface area = 18 in2 C. surface area = 180 in2 D. surface area = 1144 in2 Example 3

B. Jenny has some boxes for shipping merchandise B. Jenny has some boxes for shipping merchandise. Each box is in the shape of a rectangular prism with a length of 18 inches, a width of 14 inches, and a height of 10 inches. Find the volume of the box. A. volume = 1144 in3 B. volume = 14 in3 C. volume = 2520 in3 D. volume = 3600 in3 Example 3

End of the Lesson