1. AREAS AND VOLUMES

2. 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

3. Areas by the method of coordinates:x y
Y1x x1y2
Y2x x2y3
Y3x x3y4
Y4x x4y5
Y5x 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) )

4. x y x y

5. Areas of irregular figures:
Trapezoidal rule:

6. Simpson’s one-third rule:

7. 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.

8. Average End Area Method (AEA):

10. Volume By Prismoidal Formula:

11. Volume From Contour Maps
The figure

12. 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