Same Area Square and Triangle
By Page Bird
Given a triangle, how do you construct a square with the same area using dynamic geometry software?
There are several steps to this problem.
Given the triangle below, I want to be able to construct a square with the same area:
I can easily construct a rectangle with the same area by following these steps:
The original triangle and the resulting rectangle have the same area.
So I have a rectangle with the desired area but not a square.
The area of the rectangle is given by A = w * h.
So the sides of the square that we need must have a length of
In order to construct a length of
I am going to use the geometric mean. First I will construct a circle with a diameter of w+h:
The center of this circle is V.
Next I will construct a perpendicular segment where w and h meet:
The algebraic relationship between the segments is:
Since I have constructed a length representing the square root of wh, then I can construct a square with a side of this length:
The area of the square above is wh which is the same as the rectangle and the triangle.
Check out a dynamic GSP of this. You can play around with this and see for yourself that the triangle, rectangle, and square have the same area.