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Sect 12.8: Contrasting and Relating the Perimeter and Area of Shapes

Sect 12.8: Contrasting and Relating the Perimeter and Area of Shapes. Comparing Area and Perimeter. Recall: The perimeter of a polygon is the distance around the shape, or the sum of the side lengths Ex 1: The Aztec diamond from Exam 2. Comparing Area and Perimeter.

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Sect 12.8: Contrasting and Relating the Perimeter and Area of Shapes

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  1. Sect 12.8: Contrasting and Relating the Perimeter and Area of Shapes

  2. Comparing Area and Perimeter • Recall: The perimeter of a polygon is the distance around the shape, or the sum of the side lengths • Ex 1:The Aztec diamond from Exam 2

  3. Comparing Area and Perimeter • Recall: The perimeter of a polygon is the distance around the shape, or the sum of the side lengths • Ex 1:The Aztec diamond from Exam 2

  4. Adding Area to the Aztec Diamond • Ex 2: The Aztec diamond with 1 block added to the side

  5. Adding Area to the Aztec Diamond • Ex 3: The Aztec diamond with 1 block added to the bottom right corner.

  6. Conclusions from those examples • Even though adding area usually adds to the perimeter, it doesn’t always do so. • Area and perimeter aren’t directly related by an equation for general two-dimensional shapes. (They are for circle)

  7. See Activity 12 S

  8. Area when perimeter if fixed • For any fixed perimeter of P units, there is a rectangle with area A for any number A such that • What is the biggest shape that can be made by 20 inches of string? The circle of radius inches is the largest. • Fixing perimeter constrains the possible area of a shape.

  9. Sect 12.9: The Pythagorean Theorem

  10. The Pythagorean Theorem: For a right triangle with a hypotenuse of length c with the other two sides having lengths a and b, the following equation relates the 3 side lengths:

  11. Example Problem • Ex 1: Find the length of a basketball shot from the opposite corner of the court if the court is 50 feet wide and 94 feet long.

  12. Pythagorean Triples • Pythagorean Triples: whole number values (a, b, c) that are solutions to the equation • Ex’s: (3, 4, 5) (6, 8, 10) (9, 12, 15) ….. (5, 12, 13) (10, 24, 26) …. (8, 15, 17) (7, 24, 25)

  13. Proofs of the Pythagorean Theorem • There are literally dozens of proofs of this, dozens! • http://themetapicture.com/media/why-couldnt-i-have-been-shown-this-in-maths-class.gif • http://www.cut-the-knot.org/pythagoras/index.shtml

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