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Mastering Geometry: Area of Shapes and Solids Fundamentals

Learn how to calculate the area of various shapes and solids, including composite figures, circles, prisms, pyramids, cylinders, cones, cubes, spheres, and more. Explore Cavalieri's Principle and volumes of different geometric figures.

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Mastering Geometry: Area of Shapes and Solids Fundamentals

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  1. Chapter 9 and 10Journal Marcela Janssen

  2. areas

  3. Areas • Trapezoid(base1 x base2)h 2 • Kite(½ diagonal2) diagonal 1 • Rhombus (½ diagonal2) diagonal 1 • Anypolygonwithany # of sidesArea = (½ sa) n • Squarebase x height • Rectanglebase x height • Trianglebase x height 2 • Parallelogrambase x height

  4. Examples Area 9m 9m x 3m 27m2 3m 6mm 6mm x 4mm 4mm 2 24/2 = 12mm2

  5. Composite figures

  6. Composite figures Composite Figure: A plane figure made up of triangles, rectangles, trapezoids, circles, orother simple shapesor a three dimensional-figure made up of prisms, cones, pyramids, cylinders and other simple three-dimensional figures. Plane figure Tridimensional composite figure

  7. Tofindthearea of a composite figure: • Divide tha figure into simple shapes • Findtheareas of the simple shapes • Addall of theareas of the simple shapestogetthearea of thewholecomposite figure

  8. Example 6cm Ill 7 cm 1 cm 2 cm 12 cm (10 x 6)2 2 120 2 60 cm 12x 2 24 60 + 24 84 cm 2

  9. Areas of circles

  10. Areas of Circles • Tofindthearea of a circlejust use theequation: Area = π r 2

  11. SOLIDS

  12. Solids A solidis a three-dimensional figure. Sphere Triangular prism Rectangular ppyramid

  13. PRISM

  14. Prisms Prism: isformedby 2 ll congruentpolygonal faces called bases by faces that are parallelogram. Differencebewteen a prism and a pyramid:

  15. Whatdoesit look like?

  16. Tofindthesurfacearea of a prism: Surface Area = (perimeter of base) L + 2(Area of base) Example: Surface A. = (16m) 7 + 2(24m2) 112 + 48 160 m2

  17. A net is a diagram of thesurfaces of a tridimensiitionalobject,

  18. AREA OF CYLINDER

  19. Cylinders Isformedbytwoparallelcongruent circular bases and a curvedsurfacethatconnectsthe bases. Tofindthesurfacearea: SurfaceArea = 2(π r 2) + (2π r)h

  20. Examples

  21. AREA OF PYRAMID NOT EXAMPLES

  22. Pyramid Tofindthe total surfacearea: ½ pl + b L= lenght of the lateral face A= area of the base P= perimeteropfthe base

  23. AREA OF CONE NOT EXAMPLES

  24. Cone Tofindthesurfacearea of a cone: π r√r2 +h2 R= radius H = height

  25. AREA OF CUBE

  26. Cube A cube is a squareprismwith 6 congruent faces. Tofindthesurfacearea: 6 a 2 A = lenght of edges

  27. Example 1 Surface Area = 6(5 in)2 = 6(25) = 150 in2

  28. Example 2 Given that the height of a cube is 5 ft 3 in what is the surface area that it has? Surface Area = 6(5 ft 3 in)2 = 6(63 in) = 6(63) = 378 in2 = 31.5 ft2

  29. Example 3 Howmuchisthesurfacearea of thisrubiks cube? Surface Area = 6(8 in)2 = 6(64) = 384 in2

  30. CAVALIERI’S PRINCIPLE

  31. Cavalieri’sPrinciple • Iftwothree-dimensional figures havethesame base area, and sameheight, theywillhavethesamevolume.

  32. VOLUMES

  33. Volume • Prism • Cylinder πr2 • Pyramid 1/3 bh Cone 1/3 π r2 h

  34. SPHERES

  35. Spheres Sphere: A tridimensional solidcreatedbyallpointsequidistant (radius) fromthe center point. Hemisphere: Half of a sphere Great Circle: Any line drawnaroudthespherethatcutsitintotwohemisphere (equator)

  36. Surfacearea of a sphere: 4 π r2 Example: r= 8.5 4π 8.52 4π 17 Volume of a sphere: 4/3 πr3 How many water is needed to fill this sphere with water with a radius of 8.5? 4/3 πr3 4/3 π 8.53 86 mm

  37. TO BE GRADED: • SPHERES • PRISMS • AREA OF A CUBE

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