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**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.

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**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. r h r h 2πr

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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.

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**Problem #1 Find the dimensions of 2 other boxes with the same volume.**

Make up 2 dimensions, say 4 x 10. Then solve: 4 x 10 x H = 240. One example is 4 x 10 x 6. Write 2 sentences describing your findings. Anything along the lines that figures that have the same surface area may not have the same volume, and vice versa.

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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.

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**Sketch a cylinder with radius r and height h**

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.

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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.

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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 un SA: 144 • π un2. Cylinder B: radius 12 in., height 3 in. Vol.: 432 • π un SA: 360 • π un2. Cylinder C: radius 3 in., height 12 in. Vol.: 108 • π un 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.

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**Summary Prisms Cylinders Pyramids**

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, see

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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.

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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.

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**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). b H s

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**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 H h a c b

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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!!!

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**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.

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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?

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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?

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**Make them equal capacities**

Suppose I have two boxes: Where should I make a cut so that the boxes have the same capacities? 2.5” 12.5” 8” 4.5” 2.1” 8”

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**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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**What percent of the quilt block is shaded pink?**

Explain your reasoning. If this pattern is found on all 6 sides of a cube, find the percent of surface area that is pink.

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