Transcript Geometry Project

Group 6 Period 5
Problems 34-38
Mac Smith, Jacob Sweeny Jack
McBride
Prism- A 3 dimensional polygon
Base- Congruent polygons lying on plains
Altitude (Prism)- A segment joining the two base planes, it is perpendicular to both
Lateral Faces- The faces of a prism that are not its bases
Lateral Edges- Parallel edges that connect the bases
Right Prism- A prism whose lateral faces are rectangles
Oblique Prism- Any other prism^
Lateral Area (L.A.)- The sum of the areas of a prism’s lateral faces
Total Area (T.A.)- The sum of the areas of all of a prism’s faces
Cube- A rectangular solid with square faces
Pyramid- A 3D shape whose lateral edges meet at one point instead of the corners of
a base
Vertex- The point where the lateral edges of a pyramid meet
Altitude (Pyramid)- The segment from the vertex that is perpendicular to the base
Regular Pyramids- Pyramids with the following properties:
1) The base is a regular polygon
2) All lateral edges are congruent
3) All lateral faces are congruent isosceles triangles
Slant Height- the height of a lateral face of a pyramid
Cylinder- A prism whose bases are circles
Right Cylinder- A cylinder whose segment joining the centers of the bases is an
altitude
Radius (Cylinder)- the radius of the base
Cone- a pyramid whose bases are circles
Similar Solids- solids with the same shape, but not necessarily same size
Formulas
• The formulas listen on the following slides are needed to
successfully complete all problems related to Chapter 12: Areas
and Volumes of Solids
– NOTE: In all of the following formulas,…
• The term “B” denotes the area of a base of the solid figure that the formula
pertains to
• The term “V” denotes the volume of the solid figure that the formula pertains to
• The term “p” denotes the perimeter of the base of the solid figure that the
formula pertains to
• The term “l” denotes the slant height of the pyramid or cone that the formula
pertains to
• The term “s” denotes the side length of the base of the prism or pyramid that
the formula pertains to
• The term “r” denotes the radius of the base of the cylinder or cone that the
formula pertains to
• The term “h” denotes the height of the prism, pyramid, cylinder or cone that
the formula pertains to
• The term “S.A.” denotes the surface area (equal to the total area) of the prism,
pyramid, cylinder, cone or sphere that the formula pertains to
• The term “L.A.” denotes the lateral area of the prism, pyramid, cylinder, cone
or sphere that the formula pertains to
• The term “T.A.” denotes the total area (equal to the surface area) of the prism,
pyramid, cylinder, cone or sphere that the formula pertains to
Formulas
• Formulas for Prisms:
– The lateral area of a right prism equals the
perimeter of a base times the height of the
prism. (L.A. = ph)
– The total area of any prism equals the lateral
area of the prism plus two times the base
area.
(T.A. = L.A. + 2B)
– The volume of a right prism equals the area of
a base times the height of the prism.
(V = Bh)
Formulas
• Formulas for Pyramids:
– The lateral area of a regular pyramid equals half the
perimeter of the base times the slant height.
1
(L.A. = pl)
2
– One can also find the lateral area of a regular pyramid
with n lateral faces by finding the area of one lateral
face and multiplying
it by n.
1
(L.A.
= n( sl))
2
– The total area of any pyramid equals the lateral area
of the pyramid plus the base area.
(T.A. = L.A. + B)
– The volume of a regular pyramid equals one third the
area of
the base times the height of the pyramid.
1
(V = Bh)
3
Formulas
• Formulas for Cylinders:
– The lateral area of a right cylinder equals the
circumference of a base times the height of the
cylinder.
(L.A. = 2πrh)
– The total area of any cylinder equals the lateral
area of the prism plus two times the base area.
(T.A. = L.A. + 2B)
– The volume of a right cylinder equals the area of
a base times the height of the cylinder.
(V = Bh, or V = πr²h)
Formulas
• Formulas for Cones:
– The lateral area of a right cone equals half the
circumference of the base times the slant
1
height.(L.A.
= ∙ 2πr ∙ l, or L.A. = πrl)
2
– The total area of any cone equals the lateral
area of the cone plus the base area.
(T.A. = L.A. + B)
– The volume of a right cone equals one third
the area of the base times the height of the
cone.13 (V = πr²h)
Formulas
• Formulas for Spheres:
– The surface area of a sphere equals 4π times
the square of the radius.
(S.A. = 4πr²)
– The surface area of a sphere is the same as
the sphere’s total area and lateral area.
(S.A. = L.A. = T.A.)
4
– The volume of a sphere equals
π times the
3
cube of the radius.
4
(V = πr³)
3
Additional Theorem
• If the scale factor of two similar solids is
“a : b,” then…
1. The ratio of corresponding perimeters is “a :
b.”
2. The ratio of the base areas, of the lateral
areas, and of the total areas (or the surface
areas) is “a² : b².”
3. The ratio of the volumes is “a³ : b³.”
Problem 34
Problem 35
Problem 36
Problem 37
The NEW is DOUBLE the OLD in
Respective VOLUMES
Problem 38