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1.7: Various Formulas

1.7: Various Formulas. Objectives: To use formulas to find the perimeter, circumference, and area of squares, rectangles, triangles, and circles To identify various polyhedra and solids of revolution To use formulas to find the volume of prisms, pyramids, cylinders, cones, and spheres.

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1.7: Various Formulas

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  1. 1.7: Various Formulas Objectives: • To use formulas to find the perimeter, circumference, and area of squares, rectangles, triangles, and circles • To identify various polyhedra and solids of revolution • To use formulas to find the volume of prisms, pyramids, cylinders, cones, and spheres

  2. Vocabulary In your notebook, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

  3. Units vs. Square Units The length of something (a segment, a room, a board) is measured in units: inches, feet, centimeters, meters, etc. It’s a one-dimensional measurement.

  4. Perimeter The perimeter of a polygon is the sum of lengths of all the segments that make up the polygon. It is basically the distance around the shape. Perimeter = 5.35 + 4.82 + 6.04 + 4.30 +3.72 = 24.23 cm

  5. Let’s Have Some π The distance around a circle is its circumference. Circumference

  6. Units vs. Square Units The area of something (a polygon, a floor, a wall) is measured in square units: in2, ft2, cm2, m2, etc. It’s a two-dimensional measurement.

  7. Area The area of a plane figure is the measure, in square units, of the region enclosed by the figure. • This is simply the number of unit squares that can be arranged to completely cover the figure. • If two polygons are congruent, what do you suppose is true about their areas?

  8. Formula City (MEMORIZE THESE)

  9. Example 2 Find the area and perimeter (or circumference) of each figure. P=37.4 A=74.1 C=4π OR 12.6 A= 4π OR 12.6 P=6.4 A=2.56

  10. Example 3 The recently up and running Large Hadron Collider is a circular tunnel that spans a 27 km circumference. What is the diameter of the LHC ring? 27 OR 2.2 π

  11. Example 4 If the irrigated circular field shown has an area of 804.25 m2, what is the length of the rotating sprinkler (the radius of the circle)? 16

  12. Example 5 Find the area of the blue figure shown. 642.5

  13. Example 6 Given any of the previous formulas, what would it mean to solve for a particular variable? To solve for a variable in an equation or formula means to isolate that variable on only one side of the equation: variable = everything else

  14. Example 7 • Solve C = 2πr for r. Then find the radius of a circle with a circumference of 44 in. • Solve V = (4/3)πr3 for r. Then find the radius of a sphere with a volume of 36 cm3. C = r 2π 7 = r 44 = r 2π r= 2.05 r= 8.59 r= 3V 4π r= 27 π r= 3(36) 4π √ √ √ √ 3 3 3 3

  15. Example 8 Sometimes the variable you are solving for is part of an expression. • Solve P = 2l + 2w for l. Then find the length of the rectangle whose perimeter is 30 in. and whose width is 7 in. • Solve A = (1/2)(b1 + b2)h for b1. l = P-2w 2 l = 30-2(7) 2 l = 8 2A - b2 = b1 h

  16. Example 9 Write a formula for the area of a circle in terms of its circumference. Then find the area of a circle with a circumference of 12.5 cm. A= C2 4π A= (12.5)2 4π A = 12.43 C=π2r A=πr2 r = C A= π(C)2 2π (2π)2 A= C2 4π

  17. In Glorious 3-D! Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height.

  18. Polyhedron A solid formed by polygons that enclose a single region of space is called a polyhedron.

  19. Parts of Polyhedrons • Polygonal region (flat surface) = face • Intersection of 2 faces = edge • Intersection of 3+ edges = vertex edge face vertex

  20. Example 10 • Name all of the faces • Name all of the edges • Name all of the vertices

  21. Classifying Polyhedra Polyhedrons are classified by the # of faces: [Insert Greek prefix for # of faces]-hedron

  22. Regular Polyhedron Cosmos Dodecahedron Air Octahedron A regular polyhedronis a polyhedron whose faces are regular congruent polygons. Fire Tetrahedron Earth Hexahedron Water Icosahedron

  23. Regular Polyhedron Regular polyhedra are commonly called Platonic solids.

  24. Classification of Prisms Prisms are classified by their bases.

  25. Classification of Pyramids Pyramids are also classified by their bases.

  26. Example 11 The three-dimensional figure formed by spinning a two dimensional figure around an axis is called a solid of revolution.

  27. Solids of Revolution Cylinder Cone

  28. Volume Volume is the measure of the amount of space contained in a solid, measured in cubic units. • This is simply the number of unit cubes that can be arranged to completely fill the space within a figure. Etc

  29. Exercise 12 Find the volume of the given figure in cubic units. 20

  30. More Formulas! The volume of a solid is also easily computed with a formula. What does the B represent? B = area of base

  31. Exercise 13 The previous formulas for volume all contained B, the area of the base. Rewrite the following formulas without B. • Rectangular Prism: • Cylinder: • Cone: V = (l x w) h V= πr2h V = 1/3 (πr2h)

  32. Exercise 14 Find the volume of each solid. 189 126 36π

  33. Exercise 15 The triangle shown can be rotated around the y-axis or the x-axis to make two different solids of revolution. Which solid would have the greater volume? Calculate both and compare The one with the longest radius: rotated around the y-axis.

  34. Example 16 A pipe in the shape of a cylinder with a 30-inch diameter is to go through a passageway shaped like a rectangular prism. The passageway is 3 ft high, 4 ft wide, and 6 ft long. The space around the pipe is to be filled with insulating material.

  35. Example 16 What is the volume of the insulating material? Vprism= 72 Vcylinder = 29.5 Vmaterial = 42.5 ft.3

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