Geometry Help

Hello CPhill,

i think this is a good method to check the calculation.

1.  Connect PS....this forms rectangle GPST  and triangle PES

Area  of  GPST  = GP * GT  =  5 * 7  = 35 units^2

Area triangle PES  ....[base = GT and  the height  = 2]  =

(1/2)GT *  2  =  (1/2)7 * 2  = (7/2)*2  = 7  units^2

Total area  = [35 + 7]  units^2   =  42 units^2

2.   Can be partitioned into 3 triangles and one rectangle

  3  + 2.5 + 6  + 4*8  =  43.5 units^2

3.    Area  of triangle  on left  =  (1/2)B * H  = (1/2)* 8 * 3  = 12 units^2   ....plus...

       Area  of  semicircle......radius = (1/2)√[ (2 - - 1)^2  + (3 - - 5)^2 ]  =

       (1/2)√[ (3)^2  + (8)^2 ]  =  (1/2) √73

       So .... Area  of  semicircle  = (1/2)pi (√73/2)^2 = (73 / 8) pi units^2

       Total area  = [12 + (73 / 8) pi]  units^2   ≈  40.67  units^2

4.   Area  of parallelogram  =  B * H  = 6 * 1  = 6 units^2   ....plus....

      Area  of rectangle = √[ ( - 1 - - 4 )^2  + (3  - 2)^2 ] * √[ (- 1 - 1)^2  + (3 - - 3)^2 ]  =

       √ [ ( -3)^2 + 1^2] * √[ (-2)^2  + 6^2  ]  =

       √10  * √ 40  = √400  = 20 units^2

         Total area  = [6 + 20]  units^2   =   26  units^2

5.  Area of  rectangle  + area of triangle  =   

      [6 * 7]    + (1/2)[3 * 6]  =   42 + (1/2)*18  = 42 + 9  =  51 units^2

Double-check my math.....!!!

Geometry Help

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

i = 7, b = 8,
A = i + b/2 − 1 = 10

see: https://en.wikipedia.org/wiki/Pick%27s_theorem

1. What is the area of this polygon?

\(\begin{array}{|rcll|} \hline i &=& 33 \\ b &=& 20 \\ A &=& 33 + \frac{20}{2} - 1 \\ A &=& 42\\ \hline \end{array}\)

2. What is the area of this polygon?

\(\begin{array}{|rcll|} \hline i &=& 34 \\ b &=& 21 \\ A &=& 34 + \frac{21}{2} - 1 \\ A &=& 43.5 \\ \hline \end{array}\)

4. This figure is made up of a rectangle and parallelogram.
What is the area of this figure?

\(\begin{array}{|rcll|} \hline i &=& 18 \\ b &=& 18 \\ A &=& 18 + \frac{18}{2} - 1 \\ A &=& 26\\ \hline \end{array}\)

Thanks, heureka  !!!!.....that's pretty neat....I've never heard of this before and it's not taught in our schools in the US  [ at least not that I know of ]

I assume that the drawback is if the vertices of the polygon are not all grid points???

P.S. -  I look at the proof when I'm less sober than I am now......