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Geometry

Geometry. 12.2 Pyramids. V. P. T. Q. S. R. New Vocab Words. vertex Point V is the vertex of pyramid V-PQRST. base Pentagon PQRST is the base of the pyramid. *lateral faces The lateral faces of a pyramid are triangles.

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Geometry

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  1. Geometry 12.2 Pyramids

  2. V P T Q S R New Vocab Words vertexPoint V is the vertex of pyramid V-PQRST. base Pentagon 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. height The length of the altitude is the height, h, of the pyramid.

  3. h l Regular Pyramids A *regular pyramidhas a regular polygon as its base. Regular pyramids have several important properties. • 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.

  4. 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. 4) Draw a slant height. Draw a regular hexagonal pyramid and its slant height. Draw a square pyramid and its slant height. Draw a regular triangular pyramid and its slant height.

  5. h h h h l l l l Complete the table for regular square pyramids. 10 8 62 + 102 = x2 6 6 12 3√21 √119 17 15 82 + 172 = x2 12 17 8 8 16 12 15 62 + x2 = 152 62 + 152 = x2 2√34 √353 3√29 6 6 12 12 13 5, _ , 13 52 + x2 = 122 5 5 10

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

  7. 10 6 7) Find the lateral area and total area of this regular pyramid. A = ½ a(p) TA = LA + B A = ½ 3√3(36) TA = 180 + 54√3 sq. units A = 3√3(18) 30 3√3 A = 54√3 3 6 The total area of a pyramid is its lateral area plus the area of its base. T.A. = L.A. + B That makes sense!

  8. 8 8 8 12 8 8 8 10 Find the lateral area and the total area of each regular polygon. 6) 5) 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

  9. h l 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. Must be a 30-60-90. V = 1/3 B(h) 1/3 the volume of its prism. V = 1/3 24(24)(h) 122 + x2 = (12√3)2 12√3 12√3 12√2 V = 8(24)(h) 24 V = 192(h) 12 12 V = 192(12√2) 24 V = 2304√2 units cubed 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.

  10. 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)

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