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Construction Geometry

Construction Geometry. Cones Surface Area Volume. Cones. A cone is a solid figure with a single circular base. Campbells. Geometric Solids. Geometric solids can be either “right” or “oblique”. Right solids have a vertical central axis while oblique solids (shown below) do not.

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Construction Geometry

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  1. Construction Geometry Cones Surface Area Volume

  2. Cones • A cone is a solid figure with a single circular base. Campbells

  3. Geometric Solids • Geometric solids can be either “right” or “oblique”. • Right solids have a vertical central axis while oblique solids (shown below) do not. • Our lessons will deal only with “right” solids.

  4. Cones • Right cones have a vertical central axis while oblique cones do not. H r

  5. Application • A cone has properties that are actually a combination of a circle and a right triangle.

  6. A right triangle rotated on the center point of a circle forms a cone. Application

  7. Application • The slant height (l) is used to find the surface area of a cone. l

  8. Surface Area • The formula for the surface area of a cone is found on the Math Reference Sheet. • Surface Area = πrl + πr2

  9. Surface Area l • Surface Area = πrl+ πr2 • πr2 = area of the circular base • πrl= area of cone portion r r l r

  10. Practice #1 • Determine the surface area of the cone. • 1 = πr2 = π(3)2 = 9πm2 • 1 = πrl= π(3)(15) = 45 πm2 • Calculate the sum. • SA = 54 π m2 • ≈ 169.6 m2 3 m 15 m

  11. Application • On construction jobs, most times concrete is delivered by a truck.

  12. Application • Other times carpenters must mix concrete using a concrete mixer.

  13. Application • Carpenters often calculate materials before starting a job to assure they have enough to finish without reordering. • Aggregate rock is delivered by the truckload. It is then dumped in a near-conical shape.

  14. Application • When mixing concrete, carpenters often need to calculate materials that are on site.

  15. Application

  16. Application

  17. Application • Aggregates like sand and gravel can be calculated by using the properties of a cone.

  18. Practice #2 • Find the surface area of the pile of gravel. 10’ 16’

  19. Practice #2 • For the surface area of the cone: • d = 16’ so r = 8’. • 1 = πr2 = π(8)2 = 64 πft2 • 1 = πrl= π(8)(10) = 80πft2 • Calculate the sum. • SA = 144π ft2 • ≈ 452.4 ft2 10’ 16’

  20. Application • The vertical height is needed to find the volume. l h r

  21. Volume • The formula for the volume of a cone is found on the Math Reference Sheet. • V = ⅓ πr2 h

  22. Volume • Compare the formulas for the volume of a cylinder and cone. V = πr2 h V = ⅓ πr2 h

  23. Volume • The volume of 1 cylinder = the volume of 3 cones of the same diameter and height. V = πr2 h V = ⅓ πr2 h

  24. V = πr2 h V = ⅓ πr2 h Volume 1/3 1/3 1/3 1 cylinder 3 cones

  25. 9 ft 11 ft Practice #3 • Determine the volume of the cone shaped pile.

  26. 9 ft 11 ft Practice #3 • V = ⅓ πr2 h • V = ⅓ π(11 x 11)9 • V = 363 π ft3 • V ≈ 1140.4 ft3

  27. 10 ft 8 ft Practice #4 • The vertical height can be difficult to determine for a gravel pile; use the slant height to determine the volume.

  28. 10 ft 8 ft Practice #4 • V = ⅓ πr2 h - Use the Pythagorean Theorem to determine the vertical height. • 82 + h2 = 102 64+ h2 = 100 h2 = 36h= 6’ • V = ⅓ π(8x8)6 • V = 128 π ft3 • V ≈ 402.1 ft3 6 ft

  29. 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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