CHAPTER XII
Find the Lateral Area of a Regular Pyramid with Any Number of Lateral Faces
1) Find the area of one lateral face and multiply it by the number of lateral faces. 2) Use the formula LA = ½pl, stated as the next theorem.
Properties: Regular Pyramids
1) The base is a regular polygon. 2) All lateral edges are congruent. 3) All lateral faces are congruent isosceles triangles. 4) The height of a lateral face is called the slant height of the pyramid (l). 5) The altitude meets the base at its center, O.
Definition: Altitude of a Prism
An altitude of a prism is a segment joining the two base planes and perpendicular to both. The length of an altitude is the height of the prism.
Definition: Oblique Prism
If the lateral faces of a prism are not rectangles, then the prism is an oblique prism.
Defintion: Right Prism
If the lateral faces of a prism are rectangles, then the prism is a right prism.
Definition: Bases of a Prism
The bases of a prism are the two faces that are congruent polygons lying in parallel planes.
Theorem: The Lateral Area of a Cone
The lateral area of a cone equals half the circumference of the base times the slant height. (LA = πrl)
Theorem: The Lateral Area of a Cylinder
The lateral area of a cylinder equals the circumference of a base times the height of the cylinder. (LA = 2πrh)
Theorem: Lateral Area of a Regular Pyramid
The lateral area of a regular pyramid equals half the perimeter of the base times the slant height. (LA = ½pl)
Theorem: Lateral Area of a Right Prism
The lateral area of a right prism equals the perimeter of a base times the height of the prism. (LA = ph)
Definition: Lateral Edges of a Prism
The lateral edges of a of a prism are the parallel segments where adjacent lateral faces intersect.
Definition: Lateral Faces of a Prism
The lateral faces of a prism are the sides of the prism that are not bases.
Theorem: The Volume of a Cone
The volume of a cone equals one third the area of the base times the height of the cone. (V = 1/3πr²h)
Theorem: The Volume of a Cylinder
The volume of a cylinder equals the area of a base times the height of the cylinder. (V = πr²h)
Theorem: Volume of a Pyramid
The volume of a pyramid equals one third the area of the base times the height of the pyramid. (V = 1/3Bh)
Theorem: Volume of a Right Prism
The volume of a right prism equals the area of a base times the height of the prism. (V = Bh)
Equation: Total Area
Total Area = Lateral Area + 2 * The Base