Area of a Lawn

Date: 09/04/2002 at 10:55:31
From: Randy Marshburn
Subject: Figuring square yards of grass for a lawn

I have two areas of lawn that I want to sod with new grass, and I 
don't know how much sod to buy. One lawn is 14 yds wide at one end and 
7 yds wide at the other end. The lengths are 16 yds and 20 yds. The 
other lawn is 30 yds long by 32 yds long by 5 yds wide by 10 yds wide.

The sod comes in squares. I can easily cut the sod with a shovel. Let 
me try to better describe the lawn(s). If you draw on paper the 
following. Top of the lawn is 14 yds across. The bottom side is 7 yds 
across. The left side runs 16 yds and the right side runs 22 yds. On 
paper the diagonal from the top right corner down to the bottom left 
corner is approx 20 yds. From the top left corner down to the bottom 
right corner is approx 22 yds. On the other lawn, the top of the lawn 
is 10 yds across. The bottom side is 5 yds across. The left side runs 
30 yds and the right side runs 32 yds. The diagonal from the top right 
corner down to the bottom left is approx 31 yds. From the top left 
corner down to the bottom right is approx 33 yds.

Date: 09/06/2002 at 09:09:18
From: Doctor Jeremiah
Subject: Re: Figuring square yards of grass for a lawn

Hi Randy,

Let's calculate the area, and after that we can calculate the number 
of squares.

The area of a triangle can be found like this:

Assume that the lawn can be cut along a diagonal to make a triangle.  
Then we can use Heron's formula to find the area

  +-----------------a-----------------+
   \                               +
    \                           +
     \                       +
      \                   +
       b               c
        \           +
         \       +
          \   +
           +

First calculate the intermediate value: s = (a+b+c)/2
Then the area is:  Area = sqrt[s(s-a)(s-b)(s-c)] where sqrt[x] isthe 
square root of x.

So for your first lawn we will work on the top left triangle and then 
the lower right triangle.

   The top of the lawn is 14 yds (call that value 'a').
   The left side runs 16 yds (call this value 'b').
   The third side of the triangle is the diagonal.
   The diagonal from the top right corner down to the bottom left   
     corner is approx 20 yds (call this 'c').

Then s = (a+b+c)/2 = (14+16+20)/2 = 50/2 = 25 square yards, and the
area of this triangle is sqrt[s(s-a)(s-b)(s-c)]. If we plug in a, b, 
c, and s we get:
                             sqrt[s(s-a)(s-b)(s-c)]
                             sqrt[25(25-14)(25-16)(25-20)]
                             sqrt[25(11)(9)(5)]
                             sqrt[12375]
   upper-left triangle area: 111.24 square yards

Now we do the other triangle that makes up the lawn:

   The bottom side is 7 yds across (call that 'a').
   The right side runs 22 yds (call this 'b').
   The diagonal from the top right corner down to the bottom left 
     corner is approx 20 yds (this is 'c').

Then s = (a+b+c)/2 = (7+22+20)/2 = 49/2 = 24.5 square yards, and the 
area of this triangle is sqrt[s(s-a)(s-b)(s-c)]. If we plug in a, b, 
c, and s we get:
                             sqrt[s(s-a)(s-b)(s-c)]
                             sqrt[24.5(24.5-7)(24.5-22)(24.5-20)]
                             sqrt[24.5(17.5)(2.5)(4.5)]
                             sqrt[4823.4375]
  lower-right triangle area: 69.45 square yards

This means the total area of the lawn is the sum of the two triangular 
areas, a total of 180.69 square yards. If 1 square of sod is 1 square 
yard, then you need 181 squares of sod (because you can't buy 0.69 
squares of sod).

The other lawn would be done the same way. And in general the area of 
anything can be calculated if you can break it up into triangles.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/