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### 10-2_math113

Course: MATH 113, Winter 2009
School: Walla Walla University
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Word Count: 1054

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and Perimeter Area Area Arc Length of Circular Sectors Conclusion MATH 113 Section 10.2: Area and Perimeter Prof. Jonathan Duncan Walla Walla University Winter Quarter, 2008 Perimeter Area Area and Arc Length of Circular Sectors Conclusion Outline 1 Perimeter Area Area and Arc Length of Circular Sectors Conclusion 2 3 4 Perimeter Area Area and Arc Length of Circular Sectors Conclusion Perimeter...

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and Perimeter Area Area Arc Length of Circular Sectors Conclusion MATH 113 Section 10.2: Area and Perimeter Prof. Jonathan Duncan Walla Walla University Winter Quarter, 2008 Perimeter Area Area and Arc Length of Circular Sectors Conclusion Outline 1 Perimeter Area Area and Arc Length of Circular Sectors Conclusion 2 3 4 Perimeter Area Area and Arc Length of Circular Sectors Conclusion Perimeter Defined In this section we will examine two measurements common to two-dimensional figures. Perimeter Perimeter is the distance around an object. In a polygon, this can be determined by adding the length of the sides. Example Find a formula for the perimeter of each figure: 1 2 3 A square of side length s. An isosceles triangle with base length b and side length s. A rectangle of width w and height h. Perimeter Area Area and Arc Length of Circular Sectors Conclusion Perimeter and the Pythagorean Theorem One of the most well-known theorems of elementary geometry has to do with side lengths (perimeter) for certain types of triangles. Pythagorean Theorem In a right triangle as shown to the right, a2 + b 2 = c 2 Proof of the Pythagorean Theorem b a c a b c b a c b c c a b a Perimeter Area Area and Arc Length of Circular Sectors Conclusion A Pythagorean Example While it may seem like the pythagorean theorem has limited applicability, it can be useful in many different situations. Example Use the pythagorean theorem to determine the length of the lake pictured below. B A x D C Perimeter Area Area and Arc Length of Circular Sectors Conclusion Perimeter on Geoboards The Pythagorean theorem is also useful for finding the perimeter of polygons as they can often be broken into right triangles. Example Find the perimeter if each geoboard figure below. Perimeter Area Area and Arc Length of Circular Sectors Conclusion Definition of Area While perimeter is a measure of length around a figure, area is the measure of the enclosed region of the figure. Area Area is a physical quantity expressing the size of a part of a surface. Example Find a formula for the area of each figure: 1 2 3 A square of side length s. An isosceles triangle with base length b and side length s. A rectangle of width w and height h. Perimeter Area Area and Arc Length of Circular Sectors Conclusion Area of Irregular Shapes Area formulas for polygons are relatively simple as we can break these figures down into squares or triangles. How do we find the area of irregularly shaped figures? Example How could one measure this irregularly shape? 1 2 Use a grid or graph paper. Use triangles and rectangles to estimate. Use trapezoids to estimate. Find the area of a rectangle with the same perimeter. Weigh it. 3 4 5 Be Careful! Does number 4 above really work? Perimeter Area Area and Arc Length of Circular Sectors Conclusion Relation Between Area and Perimeter While there is a relationship between area and perimeter (which we have explored in lab), figures with the same perimeter need not have the same area. Example Find the area and perimeter of rectangles with the following lengths and widths. Length 80 70 60 Width 20 30 40 Perimeter Area Area and Arc Length Circular of Sectors Conclusion Square Units It is important to remember that since area is a two dimensional measurement, the units will be different. Example How man feet are there in a square yard? Example Give the number of square yards in a rectangle that is 4 feet by 8 feet. Dimensional Analysis You can keep track of the correct units in an area measurement by multiplying the linear units together. For example, a rectangle which is w feet by l feet will have area (w l)feet feet = (w l) square feet. Perimeter Area Area and Arc Length of Circular Sectors Conclusion Circumference and Arc Length The perimeter of a circle has a special name. Circumference The circumference (perimeter) of a circle is given by the formula C = d = 2r What is ? is defined to be the ratio of the circumference of a circle to its diameter. This is constant for all circles. That is, = C . Solving d this for C yields the above equation. Example Find the circumference of a circle of radius 4 and a circle of diameter 7. Perimeter Area Area and Arc Length of Circular Sectors Conclusion An Arc Length Example Using proportions, we can find the arc length, or perimeter, of a piece of a circle. Example Find the arc length of an arc of 40 from a circle of radius 6. Solution Find the circumference of the entire circle. The first ratio is the arc length to this circumference. The second ratio is the angle measure to the entire 360 in the circle. Solving this proportion yields the arc length. Perimeter Area Area and Arc Length of Circular Sectors Conclusion How Eratosthenes Measured the Earth Around 240 B.C. a Greek mathematician named Eratosthenes calculated the circumference of the earth to within 2% accuracy. Calculating the Circumference of the Earth The following is the procedure Eratosthenes used to calculate the circumference of the earth. On a certain day the sun was directly overhead in the town ...

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