Geometry 12.2 Pyramids

New Vocab Words T S R Q P V vertexPoint V is the vertex of pyramid V-PQRST. basePentagon PQRST is the base of the pyramid. *lateral faces The lateral faces of a pyramid are triangles. The segment in which the triangles meet are the lateral edges. *altitudeThe segment from the vertex perpendicular to the plane of the base is the altitude of the pyramid. heightThe length of the altitude is the height, h, of the pyramid.

Regular Pyramids h l A *regular pyramid has a regular polygon as its base. Regular pyramids have several important properties. 1)All lateral faces are isosceles triangles. The *slant height, represented by the letter l, of the pyramid is the length of an altitude of a triangle that is a lateral face. 2) All lateral edges are congruent. 3) The altitude of the pyramid meets the base at the center of the base polygon.

Drawing a regular pyramid. 1) Draw the base, it will be a slanted version because of the perspective. 2) Find the center of the base. Draw the altitude straight up from that point. 3) Connect each corner of the base to the vertex of the pyramid. Draw a square pyramid and its slant height. Draw a regular hexagonal pyramid and its slant height. Draw a regular triangular pyramid and its slant height. 4) Draw a slant height.

Complete the table for regular square pyramids Height 815 slant height, l 1015 base edge lateral edge = x 2 2√34 h l h l h l h l = x 2 √ x 2 = √ = x 2 3√ , _, x 2 = 12 2 √119

The lateral area of a regular pyramid with n lateral faces is n times the area of one lateral face ) Find the lateral area and total area of this regular pyramid. Let’s solve #7 as we learn the formula for lateral area!! The best way to know the formulas is to understand them rather than memorize them. L.A. = nF 6 10 A = ½ b(h) A = 3(10) A = 30 LA = nF LA = 6(30) LA = 180 square units OR… The lateral area of a regular pyramid equals half the perimeter of the base times the slant height. L.A. = ½ pl Why? Imagine a curtain on the ground around the pyramid(perimeter) and you are pulling the curtain up the slant height. Since the curtain will wrap around triangles we need the ½. LA = ½ pl LA = ½ 36(10) LA = 180 square units We have 6 triangles!

7) Find the lateral area and total area of this regular pyramid. The total area of a pyramid is its lateral area plus the area of its base T.A. = L.A. + B That makes sense! A = ½ a(p) √3 A = ½ 3√3(36) A = 3√3(18) A = 54√3 TA = LA + B TA = √3 sq. units

Try #5 and #6(who can answer them on the board?) Answers: 5) LA = 260 sq. units TA = 360 sq. units 6) LA = 48√3 sq. units TA = 64√3 sq. units Find the lateral area and the total area of each regular polygon ) 6)

Let’s solve #9 as we learn the formula volume!! 9. Find the volume of a regular square pyramid with base edge 24 and lateral edge 24. Draw a square pyramid with the given dimensions. 24 The volume of a pyramid equals one third the area of the base times the height of the pyramid. Why? Because a rectangular prism with the same base and same height would be V = bh and the pyramid, believe it or not, would hold 1/3 the amount of water. h l 1/3 the volume of its prism. V = 1/3 B(h) V = 1/3 24(24)(h) V = 8(24)(h) V = 192(h) Must be a √ x 2 = (12√3) 2 12√2 V = 192(12√2) V = 2304√2 units cubed

HW P. 484 CE #2-17 all P. 485 WE #1-10 (let’s do #4 together) P. 484 CE #3-17 odd P. 485 WE #1-17 odd, #27 (with a calculator)