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Construction Geometry

Construction Geometry. Pyramids Surface Area Volume. Pyramids. Pyramids are solid figures with a polygonal base and triangular sides that meet in a single point , called a vertex. Application.

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Construction Geometry

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  1. Construction Geometry Pyramids Surface Area Volume

  2. Pyramids • Pyramids are solid figures with a polygonal base and triangular sides that meet in a single point, called a vertex.

  3. Application Contrary to popular belief, the ancient Egyptians were not the only craftsmen to build pyramids.

  4. Application • Carpenters often build pyramids even today. Octagonal Pyramids

  5. Application • A pyramid is formed when a carpenter builds a roof on a regular polygonal structure, and all edges meet at a single apex. Octagonal Pyramids

  6. Rectangular Pyramid • Carpenters most often use the rectangular or square pyramid. It has a rectangular base and a perpendicular apex.

  7. Rectangular Pyramid • The shape of some roof frames are derived from rectangular pyramids.

  8. Rectangular Pyramid • This is not an actual pyramid because it does not have a single apex, but it is derived from a pyramid.

  9. Application • This roof frame is derived from a rectangular pyramid.

  10. Pyramids • Pyramids are usually identified as ones pictured in Egyptian times, with a square base and 4 triangular sides. • But pyramids can have any polygon base, such as triangles, rectangles, and pentagons.

  11. Pyramids • These are a few examples of pyramids: • Rectangular Pyramid • Triangular Pyramid • Pentagonal Pyramid

  12. Surface Area • It’s important to know the difference between the vertical height and the slant height of pyramids.

  13. Surface Area • The vertical height is the height of the pyramid and necessary when calculating the volume.

  14. Surface Area • The slant height is the height of a triangular face and necessary when calculating the area of the face and surface area of the pyramid. 8’ 1

  15. Surface Area • There is no formula for the surface area of a pyramid on the Math Reference Sheet. • It’s easiest to calculate the areas of each face and then add them all together.

  16. Practice #1 • Determine the surface area of the square pyramid. • Find the area of the square (LxW). • Find the area of the triangles (½bh). • Calculate the sum of the areas. 15.8 m 10 m

  17. Practice #1 • Determine the area of each face. • A = 10 x 10 = 100 m2 • ’s A = (4) ½(10 x 15.8) = 2(158) = 316 m2 • Calculate the sum. • SA = 416 m2 15.8 m 10 m

  18. Practice #2 • Find the surface area of the triangular pyramid. (All faces are congruent.) 8.7’ 8.7’ 10’

  19. Practice #2 • Determine the area of the faces. • A = ½(10 x 8.7) = 43.5 ft2 • 4 ’s = 4 (43.5) = 174 ft2 • SA = 174 ft2 8.7’ 8.7’ 10’

  20. Volume • The formula for the volume of a pyramid is found on the Math Reference Sheet. • V = ⅓ Bh • B = Area of the base • h = Vertical height

  21. Volume of Pyramids • A pyramid takes up ⅓ of the space of a rectangular prism with the same base (B) and height (h). V = Bh V = ⅓ Bh

  22. Volume • The volume of 1 prism = the volume of 3 pyramids of the same base and height. V = BH V = ⅓ BH 1/3 1/3 1/3 3 Pyramids 1 Prism

  23. Practice #3 • Determine the volume of the trapezoidal pyramid. 13 m 10 m 4 m 8 m

  24. Practice #3 • V = ⅓ Bh • B = ½(b1 + b2)h = ½(13+8)4 = 42 m2 • V = ⅓ (42)10 • V = ⅓ (420 m3) • V = 140 m3 13 m 13 m 10 m 4 m 4 m 8 m 8 m

  25. Practice #4 • Determine the surface area & volume.

  26. Practice #4 • Determine the surface area & volume.

  27. Practice #4 • For the surface area, determine the area of the faces. • A = 6 x 6 = 36 m2 • ’s A = (½)(6 x 5) = 15 m2 • 4(15) = 60 m2 • Calculate the sum. • SA = 96 m2 5 m 4 m 6 m

  28. Practice #4 • V = ⅓ Bh • B = bh = (6 x 6) = 36 m2 • V = ⅓ (36)4 • V = 48 m3 5 m 4 m 6 m

  29. Practice #5 • Determine the volume of the octagonal pyramid. • Determine the volume of the • pyramid with a regular octagon base.

  30. Practice #5 5’ • V = ⅓ Bh • B = ½ ap = ½ (6)40 = 120 ft2 • V = ⅓(120)8 • V = 320 ft3 6’ 8’

  31. Practice & Assessment Materials • You are now ready for the practice problems for this lesson. • After completion and review, take the assessment for this lesson.

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