Transcript Sect 12.8: Contrasting and Relating the Perimeter and Area

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
• 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
Adding Area to the Aztec Diamond
• Ex 2: The Aztec diamond with 1 block added to the side
Adding Area to the Aztec Diamond
• Ex 3: The Aztec diamond with 1 block added to the bottom right
corner.
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)
See Activity 12 S
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
2
1
0<𝐴<
4𝑃 .
• What is the biggest shape that can be made by 20 inches of string?
The circle of radius 𝑟 = 10 𝜋 inches is the largest.
• Fixing perimeter constrains the possible area of a shape.
Sect 12.9: The Pythagorean
Theorem
• 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:
2
2
2
𝑎 +𝑏 =𝑐
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.
Pythagorean Triples
• Pythagorean Triples: whole number values (a, b, c) that are solutions
to the equation 𝑎2 + 𝑏 2 = 𝑐 2
• Ex’s:
(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)
(6, 8, 10)
(9, 12, 15) …..
(10, 24, 26) ….
Proofs of the Pythagorean Theorem
• There are literally dozens of proofs of this, dozens!
• http://themetapicture.com/media/why-couldnt-i-have-been-shownthis-in-maths-class.gif
• http://www.cut-the-knot.org/pythagoras/index.shtml