Geometry 12.3 notes
Pyramid - is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex.
Altitude - of a pyramid is the perpendicular distance between the base and the vertex.
Base edge - is the intersection of the base and any lateral face.
Lateral face - has slant height “l” - the altitude of any lateral face.
Regular Pyramid - has a regular polygon for a base and it’s altitude meets the base at the center.
Regular Polygon - recall all sides are congruent and all angles are congruent.
How do you find the surface area of a pyramid?
1. Find the area of the base and the area of all the triangles that make up the lateral faces.
2. If the pyramid has a regular base, since all the triangles would have the same measurements, you can use the following formula:
Surface Area of a Regular Pyramid: B + (1/2 )Pl
B = the area of the base
P = Perimeter of the base
l = height of the lateral face
Circular Cone, or cone, has a circular base and a vertex that is not in the same plane as the base.
Altitude - the perpendicular distance between the vertex and the base.
Slant Height - l - the altitude from the vertex of the cone to the base of the cone on the side of the cone.
Surface Area of a Right Cone:
SA = (pi) r 2 + (pi) r l
Examples:
You have a regular pyramid with the altitude of the pyramid = 321 feet. The base edge is 300 feet. What is the slant height?
You know the altitude meets at the center of the pyramid so the altitude of the pyrmid, the slant height of one of the lateral faces and 1/2 the length of the base form a right triangle.
using a2 + b2 = c2
(321)2 + (150)2 = l 2
125541 = l 2
c = 354.32 = slant height.
Example: What is the surface area of the pyrmid?
SA = B + (1/2) P l
SA = (300)(300) + (4)(300)(354.32)
SA = 515184 square feet
Example:
Given a right cone with an altitude of the cone = 4cm and the diameter = 6 cm, what is the surface area?
Again, the altitude, the radius and the slant height form a right triangle:
using a2 + b2 = c2
(4)2 + (3)2 = l 2
25 = l 2
l = 5cm = slant height.
Therefore the surface area = (pi) (3)(3) + (pi)(3)(5) = 9(pi) + 15(pi) = 24(pi) = about 75.40 square cm.
HW: pg. 738 #18 - 36 even, 50, 52, 53
