chapter 12 notes   Surface Area

 

Section 12.1  Three Dimensional Figures

Polyhedron:  a solid bounded by polygons called faces. (plural is polyhedra or polyhedrons)

Edge:  a segment formed by the intersection of two faces.

Vertex:  a point where three of more edges meet. (plural is vertices)

Regular Polyhedron:  all faces are regular polygons.

 

Types of Geometric Solids:

Prism (a polyhedron):  has 2 parallel congruent faces called bases.  All other faces are quadrilaterals.  Prisms are named by their bases. (for example triangular prism, rectangular prism, hexagonal prism)

 

 

Pyramid (a polyhedron): has one base.  All other faces are triangles that intersect at one vertex.  Pyramids are named by their bases.  (for example triangular pyramid, square pyramid, pentagonal pyramid)

Triangular Pyramid                  Square Pyramid

Pentagonal Pyramid

 

Cylinder:  has 2 parallel congruent circular faces called bases.  The lateral surface is a rolled up rectangle

  

Cone:  has 1 circular face called a base.  The lateral surface is a rolled up sector.

 

 

 

 

 

 

 

 

Sphere: essentially a ball.

 

 

           

 

Platonic Solids


Tetrahedron Tetrahedron
Cube Cube
Octahedron Octahedron
Dodecahedron Dodecahedron
Icosahedron Icosahedron

 

 

Euler’s Formula

            Faces + Vertices = Edges + 2

            OR

            F + V = E + 2

                        F = E + 2 – V

                        V = E + 2 – F

 

                        E = F + V – 2

 

Section 12.2 Nets and Surface Area

 

 

 

 

 

 

 

 

 

 

 

 

 

Section 12.3 Surface Area of Prisms 

Prism:  a polyhedron with two congruent faces (called bases) lying in parallel planes.

--the other faces (called lateral faces) are parallelograms connecting the vertices of the bases.  The parallel segments connecting these vertices are called lateral edges.

 

 


 

Right Prism:  contains lateral edges that are perpendicular to the bases.

 

Oblique Prism:  contains lateral edges that are not perpendicular to the bases.

 

 

Surface Area of a Polyhedron:  is the sum of the areas of all the faces.

 Lateral Area of a Polyhedron:  is the sum of the areas of all the lateral faces.

 

Surface Area of a Right Prism

Surface Area (S) = 2 * Base Area (B) + Perimeter of the base (P) * Height of the prism (h)

OR:  S = 2B + Ph

Lateral Area of a Right Prism

Lateral Area (L) = Perimeter of the base (P) * Hieght of the Prism (h)

Or: L = Ph

 

 

The 5 steps to calculate Surface Area of Prisms

 1.  Draw and label the base.
2.   Record the height of the prism. (h=___)
3.  Calculate the area of the base.  (B=___)
4.  Calculate the perimeter of the base.  (P=___)
5.  Plug and calculate the Surface area using S = 2*B + P*h
 

Triangular Prism

 

Rectangular Prism

 

 

Pentagonal Prism

 

 

Triangular Prism

 

 

Hexagonal Prism

 

 

 

Octagonal Prism

 

 

Trapezoidal Prism

                                                  

 

 

Section 12.4 Surface Area of Cylinders   

 

Surface Area of a Right Cylinder

Surface Area (S) = 2*Base Area (B) + Circumference (C) * Height (h)

OR: S = 2B + Ch

OR:  S = 2*Π*r^2 + 2*Π*r*h

The 5 steps to calculate Surface Area of Cylinders

 1.  Draw and label the base.
2.   Record the height of the cylinder. (h=___)
3.  Calculate the area of the base.  (B=___)
4.  Calculate the circumference of the base.  (C=___)
5.  Plug and calculate the Surface area using S = 2*B + C*h

 

 

 

 

 

 

 

 

Section 12.5 Surface Area of Pyramids

 

Surface Area of a Regular Pyramid

Surface Area (S) = Base Area (B) + ˝ * Perimeter (P) * Slant Height (l)

OR: S = B + ˝ Pl

The 5 steps to calculate Surface Area of Pyramids

1.  Draw and label the base. (All Bases are Regular)
2.  Calculate (or record) the slant height of the pyramid. (l=___)
3.  Calculate the area of the base.  (B=___)
4.  Calculate the perimeter of the base.  (P=___)
5.  Plug and calculate the Surface area using S = B + 1/2*P*l

 

 

Square Based Pyramid

 

 

Pentagonal Based Pyramid

 

 

Triangular Based Pyramid

 

 

Hexagonal Based Pyramid

 

 

 

 

 

 

 

Section 12.6 Surface Area of Cones

 

Surface Area of a Right Cone

Surface Area (S) = Base Area (B) + 1/2 * Circumference (C) * Slant Height (l)

OR: S = B + 1/2 Cl

 

The 5 steps to calculate Surface Area of Cones

1.  Draw and label the base.
2.  Calculate the slant height of the cone. (l=___)
3.  Calculate the area of the base.  (B=___)
4.  Calculate the circumference of the base.  (C=___)
5.  Plug and calculate the Surface area using S =B + 1/2*C*l

                

 

 Surface Area and Volume of Spheres

 

 

 

 

Sphere with a Circumference = 60

 

 

 

 

Hemisphere with a radius = 12

 

 

 

 

Hemisphere with a great circle area = 800

 

 

 

Chapter 13 notes   Volume

 

Section 13.1 Volume of Prisms and Cylinders

 

Volume of a Prism or Cylinder

Volume of a Prism (V) = Base Area (B) * Height (h)

 

   

The 4 steps to calculate Volume of Prisms and Cylinders

1.  Draw and label the base.
2.  Record the  height of the prism or cylinder. (h=___)
3.  Calculate the area of the base.  (B=___)
4.  Plug and calculate the Volume using V = B * h 

 

                          

 

 

 

 

 

Section 13.2 Volume of Pyramids and Cones

 

Volume of a Pyramid or Cone

Volume of a Pyramid (V) =1/3 * Base Area (B) * Height (h)

 

 

The 4 steps to calculate Volume of Pyramids and Cones

1.  Draw and label the base.
2.  Calculate the  height of the pyramid or cone. (h=___)
3.  Calculate the area of the base.  (B=___)
4.  Plug and calculate the Volume using V = 1/3 * B * h

 

 

 

 

               

 

                                                              

 

 

 

   

Section 13.4 Congruent and Similar Solids

Solids are similar if and only if their corresponding measurements are all in the same ratio.

Solids are congruent if and only if their corresponding measurements are all equal.

 

Theorem 13.1:  If two solids are similar with a scale factor of a : b, then the surface areas have a ratio of a^2 : b^2 and the volumes have a ratio of a^3 : b^3

Example

If the side ratio/scale factor is 3/4,

then the surface area ratio is 9/16 (square the side ratio)

and the volume ratio is 27/64 (cube the side ratio)