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- 1. Volume & Surface Areaof Three-Dimensional Figures What’s it all about?
- 2. Volume• Volume is the three-dimensional space taken up by a polyhedron (solid figure). • It is measured in cubic units
- 3. Volume FormulasThere are two general formulas we use for the two basic shapes of three-dimensional figures: PRISMS & PYRAMIDSPrisms go straight up Pyramids come to afrom their Base to a point
- 4. Volume Formulas for Prisms • In general, the Volume formulas for any Prism is V = Bh where B is the area of the BaseRectangular Base Round Base A = lw A = r 2
- 5. Let’s try oneV = Bh or lwh V = (8)(3)(4)V = 96 ft cubed
- 6. Now try this one Find the volume of a rectangular solid (prism) shedLength: 5.5 feet Width: 7 feet Height: 9 feet If the grain to be stored in this shed sells for $ 15 per cubic foot, how much can the whole shed sell for?
- 7. Answer• A = lwh• A = (5.5)(7)(9)• A = 346.5 cubic feet• 346.5 x $15 = $5197.50
- 8. Cylinders work the same way V = Bh or r h 2 Area of the Circular Base is = (3.14) (5)^2 And then times the height V = (78.5)(9) = 706.5 cubic ins
- 9. Again, you try one1. Find the volume of a cylinder (circular prism) with the following dimensions: Radius of the circular base is 4 feet, height is 10 feet. How much grain can it hold?2. What if a supporting pole with a radius of one foot ran down the center of this flat silo? Then how much grain could it hold?
- 10. Answer1) Volume of the original cylinder V = (3.14)(4)^2 (10) = 502.4 cubic feet2) Volume of the pole V = 3.14 (1)^2 (10) = 31.4 cubic feet3) New Volume 502. 4 – 31.4 = 471 cubic feet
- 11. Now let’s look at the Volume of Pyramids (& cones) Their general formula is: V = 1/3 BhWe can take the previous examples and just divide their volume by 3. Given the sameheight and Base, the volume of pyramids and cones are 1/3 the correspondingly sized prisms.V = (1/3)(3.14)(3)^2(4)V = 37.68 cubic meters
- 12. Surface Area Surface Area is two-dimensional.It is the amount of coverage on the outside of the figure. It includes the top, bottom, and lateral sides.
- 13. Rectangular Solids are easy Add the surface areas of all six sides of this prism to get its total surface area.Top & Bottom: A = (8)(3) = 24*2 = 48 square feetFront & Back: A = (8)(4) = 32*2 = 64 square feetLeft & Right: A = (4)(3) = 12*2 = 24 square feet48+64+24 = 136 square feet total surface area
- 14. Surface Area of Square PyramidsFor any three-dimensional figure, study the shapes that make up the top, bottom & sides. Calculate the areas for each of these shapes separately and add them together at the end. What shapes do you see? How many?Side of Square base: 5 Slant height of each triangle: 8Area of Base = 25Area of triangle = (1/2)(5)(8) = 20Area of 4 lateral sides = 80Total Surface Area = 80+25 = 105 sq ins
- 15. And now the Surface Area of the CylinderCalculate the top & bottom separately from the lateral side, and then add them together. Area of circular bases: 2(3.14)(r)^2 Area of the lateral side is the circumference of the circular base * height
- 16. Let’s put some numbers in For this cylinder: Radius = 6 and Height = 8Area of Top & Bottom: 2(3.14)(6)^2 = 226.08 Area of Lateral Side: (12)(3.14)(8) = 301.44 226.08 + 301.44 = 527.52 square units
- 17. Now you tryFind the surface area of a cylinder with a radius of 3.5 inches and a height of 5 inches.
- 18. AnswerArea of top & Bottom: 2(3.14)(3.5)^2 = 76.93Area of lateral side: (3.14)(7)(5) = 109.9Add: 76.93 + 109.9 = 186.83 square units
- 19. Go for this from your book
- 20. And these

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