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Applying the Surface Area Formula for Cylinders

Applying the Surface Area Formula for Cylinders
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2022-12-07 01:42
Solid GeometryGeometric Construction
This example demonstrates how to calculate the base areas, lateral areas, and surface areas of right circular cylinders by unfolding and folding nets. It involves core mathematical concepts such as the geometric properties of cylinders and the application of surface area formulas. The construction utilizes GeoGebra commands and objects like Sequence, Shuffle, Element, Zip, Textfield, If, Segment, Circle, Polygon, Corner, Rotate, Vector, Translate, Cylinder, and Midpoint. It is suitable for students in grades 6-8 learning to find the lateral and surface areas of prisms, pyramids, cylinders, and cones using formulas.

GeoGebra Commands & Parameters

  1. Sequence(5) → ListRadius_{1}
  2. Shuffle(ListRadius_{1}) → randomListRadius_{1}, {4, 5, 1, 3, 2}
  3. Element(randomListRadius_{1}, Mod[countQuestion_{1}, Length[randomListRadius_{1}]] + 1) → randomRadius
  4. Zip(If[Mod[kk, 3] ≟ 0, Sequence[1 / 3, 5, 1 / 3], If[Mod[kk, 2] ≟ 0, Sequence[1 / 2, 5, 1 / 2], Sequence[5]]], kk, randomListRadius_{1}) → ListHeights
  5. Shuffle(ListHeightsC) → randomListHeight_{1}, {0.5, 3.5, 4, 1.5, 2, 2.5, 4.5, 1, 5, 3}
  6. Textfield(input1) → InputBox1
  7. Shuffle(Sequence[0.1, 2, 0.01]) → randomListIncrement_{1}, {1.9300000000000008, 1.5700000000000005, 0.7600000000000006, 1.9600000000000009, 1.5200000000000005, 1.2200000000000002, 1.7000000000000006, 1.7500000000000007, 0.13, 1.7800000000000007, 0.3300000000000002, 0.3800000000000002, 1.07, 0.1, 1.3500000000000003, 1.2800000000000002, 1.3300000000000003, 0.7200000000000005, 0.6800000000000005, 1.3200000000000003, 1.4200000000000004, 0.9200000000000007, 0.28000000000000014, 1.1400000000000001, 1.03, 1.9800000000000009, 1.8100000000000007, 1.6500000000000006, 0.8900000000000007, 0.6400000000000005, 1.7100000000000006, 1.4900000000000004, 0.19000000000000006, 1.1500000000000001, 1.8200000000000007, 1.9700000000000009, 1.6000000000000005, 0.11, 0.2300000000000001, 0.4600000000000003, 1.3800000000000003, 0.17000000000000004, 0.5200000000000004, 0.4500000000000003, 1.7200000000000006, 0.8100000000000006, 0.42000000000000026, 1.5900000000000005, 0.39000000000000024, 0.8400000000000006, 1.3400000000000003, 1.6200000000000006, 0.5400000000000004, 1.1, 0.5100000000000003, 0.6900000000000005, 1.4300000000000004, 1.6300000000000006, 1.4500000000000004, 1.5100000000000005, 0.7300000000000005, 0.5300000000000004, 0.9600000000000007, 0.29000000000000015, 0.9100000000000007, 0.6200000000000004, 1.1700000000000002, 0.18000000000000005, 1.9900000000000009, 1.5300000000000005, 1.2300000000000002, 1.8700000000000008, 0.3400000000000002, 1.8000000000000007, 0.8800000000000007, 0.27000000000000013, 1.4400000000000004, 1.02, 0.6300000000000004, 0.21000000000000008, 1.3000000000000003, 1.9000000000000008, 0.7400000000000005, 0.7700000000000006, 0.4900000000000003, 0.40000000000000024, 0.9400000000000007, 1.9500000000000008, 0.5800000000000004, 0.2600000000000001, 1.4800000000000004, 1.12, 1.4000000000000004, 0.3700000000000002, 1.7400000000000007, 0.6000000000000004, 0.6700000000000005, 0.5700000000000004, 1.5800000000000005, 1.9100000000000008, 0.8200000000000006, 0.9700000000000008, 0.2500000000000001, 1.1300000000000001, 0.8000000000000006, 0.7800000000000006, 1.6900000000000006, 1.2700000000000002, 1.3600000000000003, 0.14, 0.20000000000000007, 0.9000000000000007, 0.41000000000000025, 1.3100000000000003, 1.7900000000000007, 0.15000000000000002, 1.2100000000000002, 0.4800000000000003, 1.2000000000000002, 0.3200000000000002, 1.04, 1.3700000000000003, 1.6600000000000006, 0.12, 1.9200000000000008, 0.7000000000000005, 1.2600000000000002, 1.2500000000000002, 1.7300000000000006, 0.7900000000000006, 1.09, 1.8300000000000007, 0.8300000000000006, 0.9500000000000007, 0.9900000000000008, 1.4100000000000004, 1.9400000000000008, 1.1800000000000002, 1.7700000000000007, 1.6100000000000005, 0.9300000000000007, 1.4600000000000004, 1.01, 1.8600000000000008, 0.7100000000000005, 1, 0.31000000000000016, 0.2400000000000001, 1.2900000000000003, 0.9800000000000008, 1.5600000000000005, 1.8400000000000007, 1.6400000000000006, 1.11, 1.5500000000000005, 1.5400000000000005, 0.5600000000000004, 1.7600000000000007, 0.6500000000000005, 1.4700000000000004, 0.6600000000000005, 0.8600000000000007, 1.6800000000000006, 0.4700000000000003, 1.8900000000000008, 1.6700000000000006, 0.30000000000000016, 0.22000000000000008, 0.8700000000000007, 1.8500000000000008, 0.3500000000000002, 1.3900000000000003, 0.7500000000000006, 0.5000000000000003, 0.43000000000000027, 1.2400000000000002, 1.1600000000000001, 0.3600000000000002, 2, 1.5000000000000004, 1.06, 0.5900000000000004, 0.4400000000000003, 0.16000000000000003, 1.1900000000000002, 0.6100000000000004, 0.5500000000000004, 1.8800000000000008, 1.08, 1.05, 0.8500000000000006}
  8. Shuffle(Sequence[15]) → randomList2, {14, 8, 9, 6, 5, 2, 15, 4, 1, 13, 12, 11, 3, 7, 10}
  9. Shuffle(Sequence[20]) → randomList1, {13, 3, 19, 1, 18, 20, 5, 10, 14, 12, 16, 11, 2, 9, 8, 4, 7, 15, 17, 6}
  10. Element(randomList2, Mod[countQuestion, 15] + 1) → randomNum2
  11. Element(randomList1, Mod[countQuestion, 20] + 1) → randomNum1
  12. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin((-θCil_{1}))), ((rCil_{1} * (1 - cos((-θCil_{1})))) * cos(ϕCil_{1})) - (hei_{1} * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos((-θCil_{1})))) * sin(ϕCil_{1}))), ((-pi), ((-hei_{1}) * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})))) → CE_{1}
  13. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin(θCil_{1})), ((rCil_{1} * (1 - cos(θCil_{1}))) * cos(ϕCil_{1})) - (hei_{1} * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos(θCil_{1}))) * sin(ϕCil_{1}))), (pi, ((-hei_{1}) * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})))) → CF_{1}
  14. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin((-θCil_{1}))), ((rCil_{1} * (1 - cos((-θCil_{1})))) * cos(ϕCil_{1})), ((rCil_{1} * (1 - cos((-θCil_{1})))) * sin(ϕCil_{1}))), ((-pi), 0, 0)) → CG_{1}
  15. If(abs(tCil_{1}) < 1, ((rCil_{1} * sin(θCil_{1})), ((rCil_{1} * (1 - cos(θCil_{1}))) * cos(ϕCil_{1})), ((rCil_{1} * (1 - cos(θCil_{1}))) * sin(ϕCil_{1}))), (pi, 0, 0)) → CH_{1}
  16. Segment(CE_{1}, CG_{1}) → mCil_{1}
  17. Segment(CF_{1}, CH_{1}) → nCil_{1}
  18. If(abs(tCil_{1}) < 1, CurveCartesian[(rCil_{1} * sin((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1}))), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * cos(ϕCil_{1})), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * sin(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1], CurveCartesian[(pi * CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu), 0, 0, CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1]) → lCil_{1}
  19. If(abs(tCil_{1}) < 1, CurveCartesian[(rCil_{1} * sin((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1}))), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * cos(ϕCil_{1})) - (hei_{1} * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * sin(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1], CurveCartesian[(pi * CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu), ((-hei_{1}) * sin(ϕCil_{1})), (hei_{1} * cos(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1]) → jCil_{1}
  20. If(abs(tCil_{1}) < 1, Surface[(rCil_{1} * sin((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1}))), ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * cos(ϕCil_{1})) - (CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv * sin(ϕCil_{1})), (CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv * cos(ϕCil_{1})) + ((rCil_{1} * (1 - cos((CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu * θCil_{1})))) * sin(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1, CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv, 0, hei_{1}], Surface[(pi * CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu), ((-CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv) * sin(ϕCil_{1})), (CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv * cos(ϕCil_{1})), CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGu, -1, 1, CLIPBOARDmagicSTRINGCLIPBOARDmagicSTRINGv, 0, hei_{1}]) → iCil_{1}
  21. Circle(BaseCenter_{1}, 1, Vector[(0, 0, 1)]) → BaseCil2
  22. Segment((0, 1, 0), (1, 1, 0)) → segRad_{1}
  23. Polygon({((-pi), 0, 0), (pi, 0, 0), (pi, (-hei_{1}), 0), ((-pi), (-hei_{1}), 0)}) → unfoldedCilinder_{1}
  24. Corner(5) → L_{1}
  25. Circle(BaseCenter_{1}, 1, Vector[(0, 0, 1)]) → BaseCil
  26. Circle(Translate[BaseCenter_{1}, Vector[(0, 0, hei_{1})]], 1, Vector[(0, 0, 1)]) → TopCilFixed
  27. Rotate((0, 0, hei_{1}), (pi / 2 * tCil_{1}), Line[(1, 0, 0), (-1, 0, 0)]) → TopPoint
  28. Vector((0, cos((tCil_{1} * pi)), (-sin((tCil_{1} * pi))))) → u
  29. Translate(TopPoint, u) → TopPointC
  30. Vector((0, sin((tCil_{1} * pi)), cos((tCil_{1} * pi)))) → v
  31. Circle(TopPointC, 1, v) → TopCil
  32. Corner(5) → A
  33. Cylinder(baseHint, heiHint) → a, topHint, b
  34. Segment(TopPoint, TopPointC) → segRadTop
  35. Midpoint(CG_{1}, CE_{1}) → labelPos
  36. Vector((1; (pi / 2 * tCil2); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2)))) → view
  37. Translate(f, Vector[(Vector[Corner[4], Corner[1]] / 500 * 16)]) → f'

AI Drawing Prompt Reference

Draw a cylinder shape. The shape should include two base circles (a bottom circle and a top circle) and a lateral surface. The lateral surface should be represented as an unfolded rectangle, which is the cylinder's side surface when flattened. The cylinder consists of two circular bases and a curved lateral surface. The lateral surface can be unfolded into a rectangle. The construction order: first draw the base and top circles, then draw the rectangle representing the unfolded lateral surface, then roll the rectangle to form the cylinder. The drawing should clearly show the geometric structure of the cylinder, including the base circles, top circle, and lateral surface rectangle.
The prompt above was compiled from the original drawing commands and is for reference only.