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Area and Volume

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  1. Area and Volume

  2. Area, Perimeter, and Circumference Day 1 • Area—number of square units enclosed • Dimensions must all be the same unit • Altitude/Height—line perpendicular to the base with triangles, parallelograms, or trapezoids

  3. Formulas • Square/Rectangle • Perimeter (distance around) P = 4 x S (square) P = 2 L + 2 W • Area A = L x W (rectangle) A = S x S (square)

  4. Formulas • Circles (1/2 diameter) • Area Radius A = ∏ r ² or ∏ x r x r Diameter • Circumference C = ∏ x d

  5. Formulas • Parallelogram A = b x h • Triangle A = ½ b x h

  6. Formulas cont… • Trapezoid A = ½ h(b1 + b2) *or you can treat the trapezoid like a composite figure by dividing it into simpler figure and finding the area then adding the totals

  7. Formulas cont… • Circles • Area -- A = ∏r² • Circumference -- C = ∏d or 2r∏

  8. Space Figures Day 2 • 3-D figures are also called space figures or solids • Vertex-where lines meet at a point • Base-flat surface on the top and bottom of the figure • Base edge-lines along the base • Lateral face-flat surface on sides of the figure • Lateral edge-lines along the sides of the figure

  9. Types of Space Figures • Prism-2 parallel bases are congruent polygons and lateral faces are rectangles sides are always rectangles base (can be any shape; same on both ends) • Pyramid-1 polygon base the lateral faces are trianglesvertex sides are always triangles base (can be any shape)

  10. Types of Space Figures • Cylinder-2 parallel bases that are congruent circlesbase (always a circle) sides are rounded • Cone-1 circular base and 1 vertex vertex base (always a circle) • Sphere-all points equal distance from center

  11. Identifying Space Figures from a Net • Net—pattern you can form into a space figure • Named for the bases • You must know what shape the bases and faces form to be able to figure out a net

  12. Nets cont… Cylinder Triangular Prism

  13. Nets cont… Cube Rectangular Pyramid

  14. Square Root and Irrational Numbers • Square Root– the inverse of squaring a number • Symbol √ • The square of an integer/number is a perfect square • On calculator: 2nd button then x² button then # then enter • Irrational Number—decimal form of a number that neither terminates or repeats • If an integer isn’t a perfect square, its square root is irrational

  15. Square Root cont… • The first 13 perfect squares are easy to memorize: 0² = 0 6² = 36 1² = 1 7²= 49 2² = 4 8² = 64 3² = 9 9² = 81 4² = 16 10²= 100 5² = 25 11² = 121 6² = 36 12²= 144 * a square is a number times itself

  16. Square Root cont… • Practice : (simplify & state whether it is rational or irrational) 1.) √64 2.) √100 3.) - √16 4.) - √121 5.) √27 6.) - √72 7.) - √50 8.) √2

  17. Volume: Prisms and Cylinders Day 3 • Volume—number of cubic units needed to fill in a 3-D figure • Cubic unit—space occupied by a cube • This is why the units are cubed

  18. Volume cont… • Rectangular Prism or Cube V = L x w x h • Cylinder V = ∏r2 x h

  19. Volume cont… Volume of Triangular V = (1/2 b x h) x h Prism Or V = b x h x h Volume of a Cone or Pyramid 2 Pyramid V = 1/3 ( L x w) x h Or V = l x w x h 3 Cone V = 1/3 (∏ r2) x h Or V = ∏r² x h 3

  20. Surface Area Day 4 • Surface Area (S.A.)—sum of the area of the bases and the lateral sides of a space figure • Draw the figure • Fill in all of the numbers for the edges • Find area of each face and add everything together • Sometimes it helps to draw a net figure then fill in the numbers

  21. Surface Area of Prisms Find each of the following areas: L x w L x h w x h Then add up all areas and multiply answer by 2 S.A. = l x w = 6 l x h = w x h = 5 sum = ? x 2 7 answer = ?

  22. Surface Area of Cylinder cont… *find area of circle then find area of rectangle add *formula for area of rectangle is (∏d x h) Area of the circle: A = ∏ r2 Ex: A = h= 11.5 cm A = ? X 2 Area of the rectangle : A = ∏ x d x h A = R = 3.5 cm A = ? Total Area = ? *

  23. Surface Area of Prism cont… EX: Area of Triangle: A = ½ b x h A = ½ (6 x 4) A = ? X 2 5 cm 6 cm 5 cm Area of Rectangle: A = l x w 6 cm 12cm A = 12 x 5 4 cm A = ? X 3 * There are 3 rectangle so multiply that area by 3 * There are 2 triangles so multiply that area by 2 Total Area = ?

  24. Surface Area of Pyramids cont… Ex: Area of Rectangle: A = l x w A = 12 x 12 A = ? 12 m 16 m Area of the Triangle: A = ½ b x h 12 m A = ½ (12 x 16) A = ? There are 4 triangles so multiply the area of the triangle by 4. Then add the area of the rectangle to the areas of the triangles. Total Area = ?

  25. Surface Area of a Cone cont… Ex: Area of the Triangle: A = ½ b x h A = ½ (8 x 10) 10 m A = ? 14 cm Area of the Circle: A = ∏ r2 A = 3.14 x 72 A = ? Add the areas of the circle and the triangle. Total Area: ?

  26. Surface Area of a Sphere cont… Ex: S.A. = 4∏ r2 d = 5 in S.A. = 4 x 3.14 x 2.52 S.A. = ?