• 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

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.

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

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.

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

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.

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 TVertex: Wno faces or edges

A. triangular pyramid

B. pentagonal prism

C. rectangular prism

D. square pyramid

A. Identify the solid.

A. cone

B. cylinder

C. pyramid

D. polyhedron

B. Identify the solid.

A. triangular prism

B. triangular pyramid

C. rectangular pyramid

D. cone

C. Identify the solid.

Platonic Solids – all of its face are regular congruent polygonsand all the edges are congruent

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

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.

• 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

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

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 = radiuss = 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.

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 Surface Area and Volume

Find the surface area and volume of the cone.

Find Surface Area and Volume

Answer: The cone has a surface area of about 75.4 cm2 and a volume of about 37.7 cm3.

r = 3, h = 4

Volume of a cone

Simplify.

Use a calculator.

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 tubewhich can be used to mail posters andarchitectural 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.

Surface Area and Volume

Surface area of a cylinder

r = 1.875 in., h = 32 in.

Answer: Mike needs about 399.1 square inches ofcardboard to make the tube.

Use a calculator.399.1

Surface Area and Volume

B. CONTAINERS Mike is creating a mailing tubewhich can be used to mail posters andarchitectural 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

Surface Area and Volume

Answer: The volume of the tube is about 353.4 cubic inches.

A. surface area = 2520 in2

B. surface area = 18 in2

C. surface area = 180 in2

D. surface area = 1144 in2

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. volume = 1144 in3

B. volume = 14 in3

C. volume = 2520 in3

D. volume = 3600 in3

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.