Quandaries and Queries


What is the total acreage of a pie-shaped piece of land that is:
146' front
85' back
246' each side

The front and back lines are fairly straight....a trapezoidal shape.





Hi Doak,

The area of a trapezoid is the average of the lengths of the parallel sides, times the distance between these two sides.

area = (x + y)/2 h

A development of this expression can be found in the answer to a previous problem.

You sent us the lengths of the two parallel sides

x = 85 feet
y = 146 feet

Since the lengths of the other two sides are equal, the symmetry in the diagram below can be used to show that triangles AFD and BCD are congruent.

Thus |CE| = |FD| and thus

146 = |CD| = |CE| + |EF| + |FD| = |CE| + 85 + |FD|


|FD| = (146 - 85)/2 = 30.5 feet

Triangle AFD is a right triangle so, using Pythagoras' Theorem

h = |AF| = sqrt(|AD|2 - |FD|2) = sqrt(2462 - 30.52) = 244.1 feet

This the area, in square feet is

 (146 + 85)/2 244.1 = 28,194.55

There are 23,560 square feet in an acre so the area, in acres is

 28,194.55/23,560 = 1.2



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