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Surface Area and Volume

Surface Area and Volume. Volume of a Cube. Take a block. Assume that each edge measures 1 unit. So, the volume of that block is 1 unit 3 . We also call this a cubic unit . Use the blocks to make 2 other cubes. How many cubic units are needed?. Volume of a cube. Answer:

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Surface Area and Volume

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  1. Surface Area and Volume

  2. Volume of a Cube • Take a block. Assume that each edge measures 1 unit. • So, the volume of that block is 1 unit3. We also call this a cubic unit. • Use the blocks to make 2 other cubes. How many cubic units are needed?

  3. Volume of a cube • Answer: • 1 cubic unit, 8 cubic units, 27 cubic units • Any “cube” will be formed with x3 blocks. • Ex: a cube with an edge that measures 13 units will have volume of 133, or 2179 cubic units.

  4. Make rectangular prisms • Make 3 different rectangular prisms, each with a base of 6 cubes. • The base must be a rectangle. Why? • The area of the base remains constant. Why? • The only thing that changes is the height. Why? • What is the volume (number of cubes) of each prism? Is this related to the L, W, and H? If so, how?

  5. Dimensions of Rectangular Prisms • Do your prisms look like this? • 3 x 2 x 1 3 x 2 x 2 • 3 x 2 x 3 3 x 2 x 4

  6. height width length Rectangular prisms • Volume: Volume is defined as area of the base multiplied by the height. • Why do we say L • W • H for a rectangular prism?

  7. Exploration 10.15 • Do 3 and 4. • Show your work and find each answer. • In 1 - 2 sentences, describe how to imagine the solution to someone who is sight-impaired or blind.

  8. Other 4-sided prisms • Suppose we had a trapezoidal prism. • Does the area of the base • height still make sense? (Hint: what is the base?)

  9. 3-sided prism • What is the base? What is the height?

  10. Other prisms • Can you find the base and height of each prism?

  11. What is a prism with a circular base? • A cylinder. • Does area of the base • height of the cylinder (prism)still make sense? • What is area of the base • height?

  12. In prisms and cylinders… • The bases are congruent. • In a prism, the faces are all rectangles. • Why aren’t the faces of a cylinder also rectangles?

  13. Surface Area of a Cube • In a cube, all six faces are congruent. • So, to find the surface area of a cube, we simply need to find the area of one face, and then multiply it by 6. • If we don’t have a cube, but we have a rectangular prism, there are still 6 faces: but they are not all congruent. • Front and back, top and bottom, right and left.

  14. Volume and Surface Area • Assume that each block has volume 1 unit3. Make 4 different polyhedra, each containing 12 cubes. • Do all four have the same volume? • Do all four have the same surface area?

  15. r h r 2πr h Formulas for surface area • Cylinder: like a prism with a circular base: • For prism: 2 • area of the base + perimeter of the base • height. • For cylinder: 2 • area of the base + circumference of the base • height: • 2 π r2 + 2π rh.

  16. Problem #1 • Suppose you have enough cardboard to make a box with dimensions 2 inches x 8 inches x 15 inches. • Vol.: 2 x 8 x 15 = 240 in3. • SA: 2(2 x 8) + 2(2 x 15) + 2(8 x 15) = 332 in2. • Find the dimensions of 2 other boxes with the same surface area. Make up 2 dimensions, say 4 x 10. Then, solve: 2(4 x 10) + 2(4 x H) + 2(10 x H) = 332. • One example is 4 x 10 x 9.

  17. Problem #2 • Sketch a rectangular prism with dimensions a x b x c. If the area of the base is doubled but the height is halved, how does the volume change? How does the surface area change? • Use real numbers. L = 5, W = 6, H = 14. • Vol.: 5 x 6 x 14 = 420 un3. • SA: 2(5 x 6) + 2(5 x 14) + 2(6 x 14) = 368 un2. • If we double the area of the base, and halve the height, then: prism might be: 5 x 12 x 7 • Vol: 5 x 12 x 7 = 420 un3 No change. • SA: 2(5 x 12) + 2(5 x 7) + 2(12 x 7) = 358 un2 Not equal.

  18. Sketch a cylinder with radius r and height h. If the radius is doubled but the height is halved, how does the volume change? How does the surface area change? • Vol: π • r2 • H vs. π • (2r)2 • H/2. π • r2 • H vs. 4 • 1/2 • π • r2. • Second cylinder holds twice as much. • SA: 2 π • r2 + 2 π • r • H vs. 2 π • (2r)2 + 2 π • r • H/2 • Second cylinder has different SA. • Write 2 sentences describing your findings. • Anything along the lines of changing these dimensions does not always yield equal vol. or SA.

  19. Problem #3 • Assume that each block has volume 1 unit3. Make 4 different polyhedra, each containing 12 cubes. • Sketch or describe each polyhedra. Then, find the volume and surface area for each. • Do all four have the same volume? • Do all four have the same surface area? • Write 2 sentences describing your findings. • Anything along the lines of the volume stays the same but the SA may change--the more the polyhedron looks like a rectangular prism, the smaller the SA.

  20. Problem #4 • Without doing any work, predict which cylinder will have the greater volume and/or greater surface area: • Cylinder A: radius 6 in., height 6 in. • Vol.: 216 • π un3. SA: 144 • π un2. • Cylinder B: radius 12 in., height 3 in. • Vol.: 432 • π un3. SA: 360 • π un2. • Cylinder C: radius 3 in., height 12 in. • Vol.: 108 • π un3. SA: 90 • π un2. • Check your predictions. Write a sentence summarizing your findings. • Anything along the lines that the size of the radius affects volume and surface area more than the height.

  21. Summary • Prisms • Volume: Area of the Base • H of prism • SA: area of 2 bases + all the faces • Cylinders • Volume: Area of Base • H of cylinder = π • r2 • H • SA: 2 • area of the base + area of the rectangle = 2 • π • r2 + 2 • π • rH • Pyramids • Volume: (1/3) • Area of the Base • H of pyramid • If we had time, we could verify this--for now, seehttp://www.wonderhowto.com/how-to/video/how-to-find-the-volume-of-pyramids-in-geometry-181525/

  22. Surface Area • Literally, the area contained by the surface of the polyhedron, cone, cylinder, or sphere. • Think of it as the amount of paint needed to paint the outside, or the exact amount of wrapping paper needed to wrap the figure.

  23. Find the surface area • Each pair has a regular, square pyramid and a triangular prism. • Step 1: Use wrapping paper, scissors, tape, etc., to exactly cover the entire polyhedron. It may be helpful to trace the faces. • Step 2: pyramid: Use a ruler to measure the edge of the base, the height of the pyramid, and the height of the height of the triangular faces. • Step 3: prism: Use a ruler to measure the edges of the triangular base, the height of the triangular base, and the height of the prism. • Do not compute! Write out the mathematics you would need to perform to determine the surface area of the pyramid and the triangular prism.

  24. s H b For the pyramid… • Area of the square base… • Area of 4 triangles (which in this case are all congruent)… • So, if the length of the side of the base is b, the height of the pyramid is H, and the height of each triangle is s, then the formula is • b2 + 4 • (1/2 • b • s).

  25. c h b H a For the triangular prism… • Area of the triangular bases… • Area of the rectangular faces… • If the sides of the triangular base are a, b, c, and the height of the triangle base is h, and the height of the prism is H, • 2 • (1/2 • a • h) + a • H + b • H + c • H = • 2 • (1/2 • a • h) + (perimeter of the base) • H

  26. The goal… • For our purposes in this class and when you teach this material… • 1. Help students to understand what all the variables in the formulas mean. • 2. Help students to understand where the formulas come from--help them to develop the formulas. • 3. Help students to apply the formulas, and know when to apply the formulas. • 4. The goal is not to memorize the formulas!!!

  27. Sketch, write the formula, substitute,and compute to find the surface area • 1. A cylinder with radius 4 cm and a height of 8 cm. • 2. A square pyramid with slant height of 8 in. and the length of the square 10 in. • A hexagonal pyramid with the area of the hexagon 40 ft.2 , a slant height of 8 ft., and the length of the side of the hexagon 6 ft. • A triangular prism with lengths of the triangle 5 m, 5 m, and 6 m, and the height of the prism 10 m.

  28. Practice Problem • Suppose you have a box that measures length 8 feet x width 10 feet x height 6 feet. • (a) Find the surface area of this box. • (b) Suppose you decrease the length by 2 feet and increase the width by 2 feet. Predict whether the surface area will change or stay the same. Explain why. • (c) Now find the new surface area. Can you explain why your initial prediction was right or wrong?

  29. Practice Problems • Suppose you have two similar cubes--one has side length 4 inches, and the other has side length 12 inches. • Find the ratio of the side lengths. • Find the ratio of the areas of the bases. • Find the ratio of the surface areas of the cubes. • Find the ratio of the volumes of the cubes. • What is the relationship between the ratios of the lengths, areas, and volumes?

  30. 8” 8” 2.5” 2.1” 12.5” 4.5” Make them equal capacities • Suppose I have two boxes: • Where should I make a cut so that the boxes have the same capacities?

  31. Compare the volumes and surface areas • A cube with side length 4 inches. • A cylinder with radius 2 inches and a height of 4 inches • In a sentence, explain what you notice.

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