MAT 142 College Mathematics Module #GMGeometryTerri L. Miller & Elizabeth E. K. Jonesrevised April 20, 2011Perimeter and AreaWe are going to start our study of geometry with two-dimensional figures. We will look atthe one-dimensional distance around the figure and the two-dimensional space covered bythe figure.The perimeter of a shape is defined as the distance around the shape. Since we usuallydiscuss the perimeter of polygons (closed plane figures whose sides are straight line segment),we are able to calculate perimeter by just adding up the lengths of each of the sides. Whenwe talk about the perimeter of a circle, we call it by the special name of circumferenceSince we dont have straight sides to add up for the circumference (perimeter) of a circle, wehave a formula for calculating this.Circumference (Perimeter) of a CircleC = 2πrr = radiusπ =the number that is approximated by 3.141593Example 1. Find the perimeter of the figure below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πL= &(9(&-/,:*(#3(%;"(<,&<="(>(9(%;"(.:$?"&(%;-%(,*(-@@&#A,$-%"/(?'(BC7874DB( 8#%(0#*#%! 7(%7,0/#%#)7#!Solution:: It is tempting to just start adding of the numbers given together, but that willnot give us the perimeter. The reason that it will not is that this figure has SIX sides andwe are only given four numbers. We must first determine the lengths of the two sides thatare not labeled before we can find the perimeter. Lets look at the figure again to find thelengths of the other sides.Since our figure has all right angles, we are able to determine the length of the sides whoselength is not currently printed. Lets start with the vertical sides. Looking at the imagebelow, we can see that the length indicated by the red bracket is the same as the length of2 Geometrythe vertical side whose length is 4 units. This means that we can calculate the length of thegreen segment by subtracting 4 from 11. This means that the green segment is 7 units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− 8!!>,#.!+#;&.!(.!0,&!/&$+0,.!'*!-//!0,&!.#4&.!-.!.,'2$! #$!0,&!*#+()&!3&/'28!! 6(77(78(8(5(;( D'2!0,-0!2&! ,-;&!-//!0,&!/&$+0,.!'*!0,&!.#4&.1 !2 &!%-$!. #57/6!%-/%(/-0&!0,&!7&)#5&0&)!36!-44#$+!0,&!/&$+0,.!0'+&0,&)!0'!+&0!8EFB@C???===+++++ !!"#$%&!7&)#5&0&).!-)&!G(.0!0,&!/&$+0,.!'*!/#$&.1!0,& !7&)#5&0&)!2'(/4!3&!EF!($#0.8!!!-)&-! In a similar manner, we can calculate the length of the other missing side using . This givesus the lengths of all the sides as shown in the figure below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− 8!!>,#.!+#;&.!(.!0,&!/&$+0,.!'*!-//!0,&!.#4&.!-.!.,'2$! #$!0,&!*#+()&!3&/'28!! 6(77(78(8(5(;( D'2!0,-0!2&! ,-;&!-//!0,&!/&$+0,.!'*!0,&!.#4&.1 !2 &!%-$!. #57/6!%-/%(/-0&!0,&!7&)#5&0&)!36!-44#$+!0,&!/&$+0,.!0'+&0,&)!0'!+&0!8EFB@C???===+++++ !!"#$%&!7&)#5&0&).!-)&!G(.0!0,&!/&$+0,.!'*!/#$&.1!0,& !7&)#5&0&)!2'(/4!3&!EF!($#0.8!!!-)&-! Now that we have all the lengths of the sides, we can simply calculate the perimeter by addingthe lengths together to get Since perimeters are just the lengths of lines, the perimeter wouldbe 50 units.The area of a shape is defined as the number of square units that cover a closed figure. Formost of the shape that we will be dealing with there is a formula for calculating the area.In some cases, our shapes will be made up of more than a single shape. In calculating thearea of such shapes, we can just add the area of each of the single shapes together.We will start with the formula for the area of a rectangle. Recall that a rectangle is aquadrilateral with opposite sides parallel and right interior angles.Area of a RectangleA = bhb = base of the rectangleh = the height of the rectangleExample 2. Find the area of the figure below.!"#$"%&'()#%"*(+"&,$"%"&(-./(0&"-( +-1"(2(#3(45( !"#!$%#!&'()&!*'!+*$%*!',%!+*,-.!'/!&#'0#*%.!1(*2!*1'3-(0#)+(')$4!/(&,% #+5!!"#!1(44!4''6 !$*!*2#!')#3-(0#)+(')$4!-(+*$)7#!$%',)-!*2#!/(&,%#!$)-!*2#!*1'3-(0#)+(')$4!+8$7#!
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