PerimeterAreaArea and Arc Length of Circular SectorsConclusionPerimeter Area Area and Arc Length of Circular Sectors ConclusionMATH 113Section 10.2: Area and PerimeterProf. Jonathan DuncanWalla Walla UniversityWinter Quarter, 2008Perimeter Area Area and Arc Length of Circular Sectors ConclusionOutline1Perimeter2Area3Area and Arc Length of Circular Sectors4ConclusionPerimeter Area Area and Arc Length of Circular Sectors ConclusionPerimeter DefinedIn this section we will examine two measurements common totwo-dimensional figures.PerimeterPerimeter is the distance around an object. In a polygon, this canbe determined by adding the length of the sides.ExampleFind a formula for the perimeter of each figure:1A square of side length s .2An isosceles triangle with base length b and side length s.3A rectangle of width w and height h.Perimeter Area Area and Arc Length of Circular Sectors ConclusionPerimeter and the Pythagorean TheoremOne of the most well-known theorems of elementary geometry hasto do with side lengths (perimeter) for certain types of triangles.Pythagorean TheoremIn a right triangle as shown tothe right,a2+ b2= c2bacProof of the Pythagorean TheoremabccccaaabbbPerimeter Area Area and Arc Length of Circular Sectors ConclusionA Pythagorean ExampleWhile it may seem like the pythagorean theorem has limitedapplicability, it can be useful in many different situations.ExampleUse the pythagorean theorem to determine the length of the lakepictured below.AxD CBPerimeter Area Area and Arc Length of Circular Sectors ConclusionPerimeter on GeoboardsThe Pythagorean theorem is also useful for finding the perimeter ofpolygons as they can often be broken into right triangles.ExampleFind the perimeter if each geoboard figure below.Perimeter Area Area and Arc Length of Circular Sectors ConclusionDefinition of AreaWhile perimeter is a measure of length around a figure, area is themeasure of the enclosed region of the figure.AreaArea is a physical quantity expressing the size of a part of a surface.ExampleFind a formula for the area of each figure:1A square of side length s .2An isosceles triangle with base length b and side length s.3A rectangle of width w and height h.Perimeter Area Area and Arc Length of Circular Sectors ConclusionArea of Irregular ShapesArea formulas for polygons are relatively simple as we can break thesefigures down into squares or triangles. How do we find the area ofirregularly shaped figures?ExampleHow could one measure this irregularly shape?1Use a grid or graph paper.2Use triangles and rectanglesto estimate.3Use trapezoids to estimate.4Find the area of a rectanglewith the same perimeter.5Weigh it.Be Careful!Does number 4 above really work?Perimeter Area Area and Arc Length of Circular Sectors ConclusionRelation Between Area and PerimeterWhile there is a relationship between area and perimeter (which wehave explored in lab), figures with the same perimeter need nothave the same area.ExampleFind the area and perimeter of rectangles with the followinglengths and widths.Length Width80 2070 3060 40Perimeter Area Area and Arc Length of Circular Sectors ConclusionSquare UnitsIt is important to remember that since area is a two dimensionalmeasurement, the units will be different.ExampleHow man feet are there in a square yard?ExampleGive the number of square yards in a rectangle that is 4 feet by 8feet.Dimensional AnalysisYou can keep track of the correct units in an area measurement bymultiplying the linear units together. For example, a rectanglewhich 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 ConclusionCircumference and Arc LengthThe perimeter of a circle has a special name.CircumferenceThe circumference (perimeter) of a circle is given by the formulaC = πd = 2πrWhat is π?π is defined to be the ratio of the circumference of a c ircle to itsdiameter. This is constant for all circles. That is, π =Cd. Solvingthis for C yields the above equation.ExampleFind the circumference of a circle of radius 4 and a circle ofdiameter 7.Perimeter Area Area and Arc Length of Circular Sectors ConclusionAn Arc Length ExampleUsing proportions, we can find the arc length, or p e rimete r, of apiece of a circle.ExampleFind the arc length of an arc of 40◦from a circle of radius 6.SolutionFind 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◦inthe circle.Solving this proportion yields the arc length.Perimeter Area Area and Arc Length of Circular Sectors ConclusionHow Eratosthenes Measured the EarthAround 240 B.C. a Gree k mathematician named Eratosthenes calculatedthe circumference of the earth to within 2% accuracy.Calculating the Circumference of the EarthThe following is the procedure Eratosthenes used to calculate thecircumference of the earth.On a certain day the sun was directly overhead in the town squareof Syene (He noticed that the sun shown straight down a well).On that same day, in the town of Alexandria 5000 stadia (489miles) to the north, the sun made an angle of 7.2◦.Using alternate interior angles, Eratosthenes determined that thecentral angle of the circular sector between the two cities is 7.2◦Using ratios, he then computed the circumference.His computed value of 24,450.00 miles is within 2% of the currentlyheld value of 24,859.82 miles.Perimeter Area Area and Arc Length of Circular Sectors ConclusionArea of a CircleThe area of a circle is a more difficult measurement to justify. Youworked on this in your lab.Area of a CircleThe area of a circle of radius r is given by A = πr2.ExampleUsing the same techniques as seen in the circumference example,find the area of a 40◦circular sector.ExampleA pizza of diameter 16” has been cut into 12 equal slices. Howmuch area is in each slice?Perimeter Area Area and Arc Length of Circular Sectors ConclusionImportant ConceptsThings to Remember from Section 10.21Computing the perimeter of polygons.2Using the pythagorean theorem to compute a perimeter.3Finding the area of a polygon.4Estimating the area of an irregular figure.5Finding area and arc length of