AREAS AND VOLUMES

Mathematical formula for area calculation: Area= ½( b×h ) Area = ½ (a ×c ×sinB ) Area =√s(s-a)(s-b)(s-c) s=(a+b+c)/2 Area = a² Area = a×b Area=π r² Area = ½ ((a+b)/2)h Area=1/4 n.a² cot(180º/n) Area=π(r2² - r1²) Area=(∆ / 360).π.r² Area = ½ r²((π∆/180)-sin∆) Area= 2/3 .b .h Area= π.a.b

Areas by the method of coordinates: x y Y1x2 x1y2 Y2x3 x2y3 Y3x4 x3y4 Y4x5 x4y5 Y5x1 x5y1 x Area = ½ ( (y1x2 + y2x3+ y3x4 + y4x5 + y5x1) – (x1y2 + x2y3 + x3y4 + x4y5 + x5y1)) = ½ ( x1(y5-y2) + x2(y1-y3) + x3(y2-y4) + x4(y3- y5) + x5( y4-y1) )

x y x y

Areas of irregular figures: Trapezoidal rule:

Simpson’s one-third rule:

Calculation of volumes: All highways and railroad construction projects involve Earthworks( Cut & Fill ) There are two methods to calculate the volumes of cut and fill.

Average End Area Method (AEA):

Volume By Prismoidal Formula:

Volume From Contour Maps The figure

Volume From Spot Levels The following figure Volume (V) = (A / 4)(∑h1 + ∑2h2 + ∑3h3 + ∑4h4 ) where A is the area of one rectangle or square and h1,h2,h3,h4 are the corner heights common to one, two, three and four rectangles respectively. Solution