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Surface Area

Surface Area. Lesson 8.7 – Surface Area HW: 8.7/1-10. Let’s start in the beginning… Before you can do surface area or volume, you have to know the following formulas. Rectangle A = lw. Triangle A = ½ bh. Circle A = π r ² C = πd. Surface Area. What does it mean to you?

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Surface Area

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  1. Surface Area Lesson 8.7 – Surface Area HW: 8.7/1-10

  2. Let’s start in the beginning… Before you can do surface area or volume, you have to know the following formulas. Rectangle A = lw Triangle A = ½ bh Circle A = π r² C = πd

  3. Surface Area • What does it mean to you? • Does it have anything to do with what is in the inside of the prism.? • Surface area is found by finding the area of all the sides and then adding those answers up. • How will the answer be labeled? • Units2 because it is area!

  4. SA of Prisms Find the SA of any prism by using the basic formula for SA: SA = 2B + LSA • LSA= Lateral Surface Area • LSA= perimeter of the base ● height of the prism • B = area of the base of the prism.

  5. B C 5 in A 6 in 4 in Rectangular Prism 6 How many faces are on here? Find the area of each of the faces. Do any of the faces have the same area? If so, which ones? Opposite faces are the same. A = 5 x 4 = 20 x 2 =40 B = 6 x 5 = 30 x 2 = 60 Find the SA C = 4 x 6 = 24 x 2 = 48 148 in2

  6. SA of a Rectangular Prism SA = 2bh+2bw+2hw Sum of the areas of all the faces. h w b

  7. Cube Are all the faces the same? YES A 6 How many faces are there? 4m Find the Surface area of one of the faces. Area of one base/face 4 x 4 = 16 * 6 96 m2 Times the number of faces SA for a Cube = 6B SA = 6 ● area of the base

  8. BACK SIDE SIDE BOTTOM FRONT TOP

  9. Triangular Prism 5 How many faces are there? 4 5 How many of each shape does it take to make this prism? 10 m 3 2 triangles and 3 rectangles = SA of a triangular prism ∙ 2= 12 2 ½ (4 ∙ 3) = 6 How many triangles were there? 5 ∙ 10 = 50 = front 4 ∙ 10 = 40 = back 3 ∙ 10 = 30 = bottom Find the area of the 3 rectangles. What is the final SA? SA = 132 m2

  10. Find the AREA of each SURFACE 1. Top or bottom triangle: A = ½ bh A = ½ (6)(6) A = 18 2. The two dark sides are the same. A = lw A = 6(9) A = 54 Example: 8mm 9mm 6 mm 6mm 3. The back rectangle is different A = lw A = 8(9) A = 72 ADD THEM ALL UP! 18 + 18 + 54 + 54 + 72 SA = 216 mm²

  11. Cylinders What does it take to make this? 2 circles and 1 rectangle= a cylinder 6 10m Formula  SA = 2B + LSA LSA is a rectangle with b = circumference of Base H = height of cylinder 2 B  B = π x 9 = 9π * 2 = 18π + LSA(C x H) SA = 18π + 60 π

  12. SURFACE AREA of a CYLINDER. Imagine that you can open up a cylinder like so: You can see that the surface is made up of two circles and a rectangle. The length of the rectangle is the same as the circumference of the circle!

  13. EXAMPLE: Round to the nearest hundredth. Top or bottom circle A = πr² A = π(3.1)² A = (9.61) π A ≈ 30.1754 Rectangle Length = Circumference C = π d C = (6.2) π C ≈ 19.468 Now add: SA = 30. 1754 + 30. 1754 + 233.62 Now the area A = lw A ≈ 19.468(12) A ≈ 233.62 SA ≈ 293.97 in²

  14. Find the surface area of the triangular prism. 9 m • Area of each triangle is 12m² (there are two of them) • Area of one of the rectangles is 63m² (7∙9) • Area of another one of the rectangles is 56m² (7∙8) • Area of the final rectangles is 21m² (7∙3)

  15. Find the surface area of the triangular prism. 9 m 12m² + 12m² + 63m² + 56m² + 21m² S.A. = 164m²

  16. Definition • Surface Area – is the total number of unit squares used to cover a 3-D surface.

  17. Find the SA of a Rectangular Solid A rectangular solid has 6 faces. They are: Front • Top • Bottom • Front • Back • Right Side • Left Side Top Right Side We can only see 3 faces at any one time. Which of the 6 sides are the same? • Top and Bottom • Front and Back • Right Side and Left Side

  18. Surface Area of a Rectangular Solid We know that Each face is a rectangle. and the Formula for finding the area of a rectangle is: A = lw Front Top Right Side Steps: Find: Area of Top Area of Front Area of Right Side Find the sum of the areas Multiply the sum by 2. The answer you get is the surface area of the rectangular solid.

  19. Find the Surface Area of the following: Find the Area of each face: 12 m Top 5 m A = 12 m x 5 m = 60 m2 Top Front Right Side 8 m Front 8 m A = 12 m x 8 m = 96 m2 5 m 12 m 12 m Right Side Sum = 60 m2 + 96 m2 + 40 m2 = 196 m2 A = 8 m x 5 m = 40 m2 8 m Multiply sum by 2 = 196 m2 x 2 = 392 m2 5 m The surface area = 392 m2

  20. Find the surface area of the rectangular prism. A. 22 in2 B. 36 in2 C. 76 in2 D. 80 in2

  21. Find the surface area of the rectangular prism. 46 in2

  22. S.A. of a Triangular Prism There is NO FORMULA! Simply find the area of each face and add them all together.

  23. Find the Surface Area Area of Top = 6 cm x 4 cm = 24 cm2 Area of Front = 14 cm x 6 cm = 84 cm2 24 m2 Area of Right Side = 14 cm x 4 cm = 56 cm2 Find the sum of the areas: 56 m2 84 m2 14 cm 24 cm2 + 84 cm2 + 56 cm2 = 164 cm2 Multiply the sum by 2: 4 cm 164 cm2 x 2 = 328 cm2 6 cm The surface area of this rectangular solid is 328 cm2.

  24. Nets • A net is all the surfaces of a rectangular solid laid out flat. Back 8 cm Top Left Side Top Right Side 5 cm 8 cm Right Side Front 8 cm Front 8 cm 5 cm 10 cm Bottom 5 cm 10 cm

  25. Find the Surface Area using nets. Top Back 8 cm Right Side Front 8 cm Left Side Top Right Side 5 cm 5 cm 8 cm 10 cm Front 8 cm Each surface is a rectangle. A = lw 80 80 5 cm Bottom Find the area of each surface. Which surfaces are the same? Find the Total Surface Area. 40 10 cm 50 50 40 What is the Surface Area of the Rectangular solid? 340 cm2

  26. SURFACE AREA • Why should you learn about surface area? • Is it something that you will ever use in everyday life? • If so, who do you know that uses it? • Have you ever had to use it outside of math?

  27. TRIANGLES You can tell the base and height of a triangle by finding the right angle:

  28. CIRCLES You must know the difference between RADIUS and DIAMETER. r d

  29. Let’s start with a rectangular prism. Surface area can be done using the formula SA = 2 lw + 2 wl + 2 lwOR Either method will gve you the same answer. you can find the area for each surface and add them up. Volume of a rectangular prism is V = lwh

  30. Example: 7 cm 4 cm 8 cm Front/back 2(8)(4) = 64 Left/right 2(4)(7) = 56 Top/bottom 2(8)(7) = 112 Add them up! SA = 232 cm² V = lwh V = 8(4)(7) V = 224 cm³

  31. To find the surface area of a triangular prism you need to be able to imagine that you can take the prism apart like so: Notice there are TWO congruent triangles and THREE rectangles. The rectangles may or may not all be the same. Find each area, then add.

  32. There is also a formula to find surface area of a cylinder. Some people find this way easier: SA = 2πrh + 2πr² SA = 2π(3.1)(12) + 2π(3.1)² SA = 2π (37.2) + 2π(9.61) SA = π(74.4) + π(19.2) SA = 233.7 + 60.4 SA = 294.1 in² The answers are REALLY close, but not exactly the same. That’s because we rounded in the problem.

  33. Find the radius and height of the cylinder. Then “Plug and Chug”… Just plug in the numbers then do the math. Remember the order of operations and you’re ready to go. The formula tells you what to do!!!! 2πrh + 2πr² means multiply 2(π)(r)(h) + 2(π)(r)(r)

  34. Volume of Prisms or Cylinders You already know how to find the volume of a rectangular prism: V = lwh The new formulas you need are: Triangular Prism V = (½ bh)(H) h = the height of the triangle and H = the height of the cylinder Cylinder V = (πr²)(H)

  35. Volume of a Triangular Prism We used this drawing for our surface area example. Now we will find the volume. V = (½ bh)(H) V = ½(6)(6)(9) V = 162 mm³ This is a right triangle, so the sides are also the base and height. Height of the prism

  36. Try one: Can you see the triangular bases? V = (½ bh)(H) V = (½)(12)(8)(18) V = 864 cm³ Notice the prism is on its side. 18 cm is the HEIGHT of the prism. Picture if you turned it upward and you can see why it’s called “height”.

  37. Volume of a Cylinder We used this drawing for our surface area example. Now we will find the volume. V = (πr²)(H) V = (π)(3.1²)(12) V = (π)(3.1)(3.1)(12) V = 396.3 in³ optional step!

  38. Try one: 10 m d = 8 m V = (πr²)(H) V = (π)(4²)(10) V = (π)(16)(10) V = 502.7 m³ Since d = 8, then r = 4 r² = 4² = 4(4) = 16

  39. Here are the formulas you will need to know: A = lw SA = 2πrh + 2πr² A = ½ bh V = (½ bh)(H) A = π r² V = (πr²)(H) C = πd and how to find the surface area of a prism by adding up the areas of all the surfaces

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