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Surface Area Formula for Cylinders

Surface Area Formula for Cylinders
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2023-08-11 03:10
Solid GeometryGeometric Construction
This example demonstrates the calculation of a cylinder's surface area by unfolding its net. Key math concepts include the lateral and total surface area formulas for cylinders. GeoGebra commands and objects used in the construction are Sequence, Shuffle, Element, Zip, Textfield, If, Segment, Circle, Polygon, Corner, Rotate, Vector, Translate, Cylinder, Midpoint, and Text. Suitable for students in grades 6-8 to learn how to use formulas to find the surface area of a cylinder.

GeoGebra Commands & Parameters

  1. Sequence(5) → ListRadius_{1}
  2. Shuffle(ListRadius_{1}) → randomListRadius_{1}, {5, 1, 4, 2, 3}
  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}, {1, 4, 5, 2, 3}
  6. Textfield(input1) → InputBox1
  7. Shuffle(Sequence[0.1, 2, 0.01]) → randomListIncrement_{1}, {0.9400000000000007, 1.11, 0.8600000000000007, 0.43000000000000027, 0.16000000000000003, 0.4600000000000003, 0.41000000000000025, 0.2600000000000001, 1.7900000000000007, 1.1300000000000001, 0.8500000000000006, 0.9300000000000007, 0.31000000000000016, 1.1900000000000002, 0.20000000000000007, 1.05, 0.6700000000000005, 1.3800000000000003, 1.4900000000000004, 1.4700000000000004, 1.8500000000000008, 0.6600000000000005, 1.9600000000000009, 0.9500000000000007, 0.15000000000000002, 1.9900000000000009, 1.9100000000000008, 0.3400000000000002, 1.6900000000000006, 1.01, 0.11, 1.04, 1.7400000000000007, 0.5200000000000004, 0.5400000000000004, 1.2000000000000002, 0.6500000000000005, 1.4400000000000004, 0.19000000000000006, 0.42000000000000026, 0.5800000000000004, 1.8000000000000007, 1.1800000000000002, 0.4400000000000003, 1.06, 1.7700000000000007, 1.7100000000000006, 0.4500000000000003, 1.3600000000000003, 1.5600000000000005, 0.3500000000000002, 0.9200000000000007, 0.8300000000000006, 0.7200000000000005, 0.7300000000000005, 0.4900000000000003, 0.28000000000000014, 0.7600000000000006, 1.6800000000000006, 1, 1.4800000000000004, 1.7200000000000006, 0.5900000000000004, 0.39000000000000024, 1.6000000000000005, 0.1, 1.9300000000000008, 0.9900000000000008, 1.1400000000000001, 1.2600000000000002, 0.18000000000000005, 0.5600000000000004, 1.5300000000000005, 1.8900000000000008, 0.6400000000000005, 1.8800000000000008, 1.9400000000000008, 1.5100000000000005, 0.6100000000000004, 1.2100000000000002, 0.29000000000000015, 1.4100000000000004, 1.2900000000000003, 1.7000000000000006, 1.03, 0.2400000000000001, 0.9100000000000007, 1.8700000000000008, 0.2300000000000001, 1.7600000000000007, 1.7800000000000007, 0.8400000000000006, 1.5800000000000005, 1.8300000000000007, 0.7800000000000006, 1.7500000000000007, 1.6500000000000006, 1.08, 0.3300000000000002, 1.4200000000000004, 0.30000000000000016, 0.8700000000000007, 0.22000000000000008, 0.4700000000000003, 0.4800000000000003, 1.9700000000000009, 1.02, 1.5700000000000005, 0.8000000000000006, 1.8400000000000007, 1.2300000000000002, 1.1, 1.5400000000000005, 1.5500000000000005, 1.9800000000000009, 1.2400000000000002, 1.6700000000000006, 1.3400000000000003, 1.8600000000000008, 0.13, 1.8200000000000007, 1.7300000000000006, 1.3100000000000003, 1.2200000000000002, 1.1700000000000002, 1.3300000000000003, 0.7100000000000005, 0.17000000000000004, 1.4300000000000004, 0.7700000000000006, 0.14, 1.9500000000000008, 1.4500000000000004, 1.3200000000000003, 0.6200000000000004, 0.9600000000000007, 0.5300000000000004, 1.6200000000000006, 0.3200000000000002, 1.2800000000000002, 1.12, 0.6300000000000004, 1.1600000000000001, 0.9800000000000008, 1.6600000000000006, 2, 1.3700000000000003, 0.3800000000000002, 0.21000000000000008, 0.7900000000000006, 1.6300000000000006, 1.3900000000000003, 1.5000000000000004, 1.5900000000000005, 1.9200000000000008, 1.2500000000000002, 0.6000000000000004, 1.1500000000000001, 0.8100000000000006, 1.3000000000000003, 0.5700000000000004, 1.6100000000000005, 0.6800000000000005, 0.8800000000000007, 0.5100000000000003, 0.2500000000000001, 0.8900000000000007, 1.2700000000000002, 1.9000000000000008, 1.4600000000000004, 1.09, 1.6400000000000006, 1.07, 0.40000000000000024, 0.3600000000000002, 0.9000000000000007, 0.27000000000000013, 0.3700000000000002, 0.6900000000000005, 0.5500000000000004, 1.4000000000000004, 1.8100000000000007, 0.7000000000000005, 1.3500000000000003, 0.7400000000000005, 0.5000000000000003, 0.8200000000000006, 1.5200000000000005, 0.7500000000000006, 0.9700000000000008, 0.12}
  8. Shuffle(Sequence[15]) → randomList2, {14, 2, 4, 9, 12, 8, 7, 15, 11, 6, 10, 1, 5, 3, 13}
  9. Shuffle(Sequence[20]) → randomList1, {13, 14, 2, 17, 1, 5, 20, 18, 4, 8, 11, 7, 12, 3, 6, 15, 16, 19, 9, 10}
  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. Midpoint(CG_{1}, CE_{1}) → labelPos
  35. Vector((1; (pi / 2 * tCil2); ((-pi)) / 2 + ((0.8 * pi / 2) * (1 - tCil2)))) → view
  36. Translate(f, Vector[(Vector[Corner[4], Corner[1]] / 500 * 16)]) → f'
  37. Rotate(TopPoint, 90°, TopPointC, v) → TopPoint'
  38. Segment(TopPoint', TopPointC) → segRadTop
  39. Text(labelRadius_{1}, TopPointC + ((0.2 * tCil2), (0.5 * tCil2), (tCil2 * hei_{1}) + 1), true, true) → text4

AI Drawing Prompt Reference

Create a 3D visualization of a cylinder together with its unfolded surface net. First draw the cylinder’s base and top circles, then form the lateral surface as a curved sheet connecting them along the cylinder height. On the lateral surface, select the boundary curve corresponding to a specific angular-to-flat unfolding position and use it to define an unfolded strip; close the strip with two height-directed line segments that act as the adjacent boundaries. Then construct the unfolded quadrilateral region using the two unfolded horizontal edges and the two vertical height edges. Finally, add a radius-related text label positioned near the unfolded net area.
The prompt above was compiled from the original drawing commands and is for reference only.