Math CentralQuandaries & Queries


Question from mike, a parent:

If the area of an equilateral triangle is 30 acres, what is the length of each side in feet or miles?


The are of a triangle is 1/2 times the base times the height. The base is the length of any side. The height of is the length of the line segment from the angle opposite that side and perpendicular to it (that side). Since the triangle is equilateral, it divides that side exactly in half to form two identical right triangles which, together, make up the whole equilateral triangle.


Each of these has one leg equal to half the length of the side of the equilateral triangles, and hypotenuse of length equal to the length of a side of the equilateral triangle. You can use Pythagoras' formula to figure out the length of the other leg, which gives you the height of the equilateral triangle.

One acre is 43560 square feet so 30 acres is 30 × 43560 square feet. Now, using 1/2 x base x height = 30 × 43560 , you can solve your problem.

Victoria and Penny

Mike wrote back

I apologize that this is the third time asking about the area of triangles. I forgot to mention
last time that it was an equilateral triangle with the area of 30 acres. Is there any way to figure
out the lengths of each side?

It is actually a piece of property and I wanted to know how much fencing I would need.
Is the equation Area = (sqrt of 3 divided by 4) x a squared?

Not a problem Mike, we want to get this right.

I agree with your expression for the area. If the length of each side is a feet then the area is

area = √3/4 a2 square feet.

Since your area is 30 acres = 30 × 43560 square feet I get

30 × 43560 = √3/4 a2

This gives me

a2 = 3017925.33 square feet


a = 1737.2 feet.


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