Lesson 12-2 Pyramids (page 482) Essential Question How can you calculate the area and volume of a pyramid?

Pyramids around the world

Pyramid in the U.S.A.

Pyramid Arena in Memphis, Tennessee Pyramid in the U.S.A. Pyramid Arena in Memphis, Tennessee

Pyramid A pyramid has only one base . Pyramids are named for their base , ie. triangular pyramid rectangular pyramid pentagonal pyramid hexagonal pyramid octagonal pyramid etc.

triangular pyramid

triangular pyramid

rectangular pyramid

rectangular pyramid

… Pyramids The lateral faces of a pyramid are triangles . The segments in which the lateral faces intersect are the lateral edges . The vertex of a pyramid is where all the lateral edges intersect. An altitude is the segment from the vertex perpendicular to the plane of the base. The height is the length of an altitude (h).

pyramid vertex lateral edge altitude lateral face base edge base

A regular pyramid has a regular polygon as its base. Regular pyramids have the following important properties: The base is a regular polygon . All lateral edges are congruent . All lateral faces are congruent isosceles triangles . The slant height ( ℓ ) is the height of a lateral face. The altitude meets the base at its center .

regular pyramid vertex slant height altitude center base edge regular polygon

regular pyramid apothem of regular polygon slant height altitude base edge radius of regular polygon

Net for a square pyramid base

Net for a square pyramid base

NOTE: The lateral faces are all congruent triangles. slant height of pyramid Perimeter of base L.A. = ½ bh = ½ pℓ

Theorem 12-3 The lateral area of a regular pyramid equals half the perimeter of a base times the slant height. L.A. = ½ pℓ

Also, if F = the area of a lateral face, then: L.A. = n F Remember the “n” is the number of sides of a polygon.

TOTAL AREA of a PYRAMID Remember a pyramid has only one base. T.A. = L.A. + B B = base area

Theorem 12-4 The volume of a pyramid equals one-third the area of a base times the height of the pyramid. V = ⅓ Bh WHY?

V = ⅓ Bh Class Demonstration: Prism and Pyramid with equal height and congruent bases.

Example: Draw a square pyramid with a height 12 Example: Draw a square pyramid with a height 12 and a slant height of 13. Then find its lateral area, total area, & volume.

Example: Draw a square pyramid with a height 12 Example: Draw a square pyramid with a height 12 and a slant height of 13. Then find its lateral area, total area, & volume. 12 13 5 s = 10

12 13 5 s = 10

12 13 5 s = 10

How can you calculate the area and volume of a pyramid? Assignment Written Exercises on pages 485 REQUIRED: 1, 7, 9, 11, 15 BONUS: Calculator Key-In on page 488 How can you calculate the area and volume of a pyramid?