help me with my sum?

Question

suppose that the 864 acre plot of land was not a rectangle but closely approximated the shape of a regular hexagon make a skech oa regular hexagon whose area is 864 acres label the plot of dimension in miles.

Answer 1

Actually, a regular hexagon can be broken into 6 equilateral triangles. (Try it!)

So the area of each of these triangles would be 864/6 = 216 acres, or 0.3375 square miles (got this value from Google).

Now, remember that each angle of an equilateral triangle is 60 degrees. If you draw a line from the peak of one of the triangles down to the base, you have 2 30-60-90 triangles. Since the line you just drew is the height of the triangle, but also the longer side of a 30-60-90 triangle, then it is √3 times the shorter side...which is 1/2 of the base.

So if the base is called s, then the area can be written as:

A = 1/2bh = 1/2s(√3/2s) = √3/4s²

0.3375 = √3/4s²
s² = 1.35/√3 ≈ 0.77942
s = √(1.35/√3) ≈ 0.88285 miles

So a side of the hexagon is about 0.88285 miles. If I didn't mess up.

(Note: since I re-read and you were looking for an answer in miles, I re-did this a little.)

Answer 2

Well i wont actually give the answer figure but i will give an idea on how to work it out. Cut two triangular portions on the sides or the top and bottom of the hexagon to obtain a square.The sum of the areas of the triangles and the square can be made equal to the area of the recangle.

Area of traingles:Base*height
Area of square:Side^2

Answer 3

Okay, first 864 acres = 1.35 sq miles.
Then i seperate the hexagon into 12 congruent triangles.
Then letting s be side, then (s/2)*(s/2)*(1/tan(360/12))* (1/2)= 1.35/12.
So, s^2 = (1.35/12)*tan30*2*2*2 = .519615 approximately.
So, s=.72084 miles approximately.

Answer 4

Each side would be approximately 0.7208428 miles in length.