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:

1. Reflect the given triangle so that a "kite" or symmetrical quadrilateral results.
2. Find the midpoints of each of the sides
3. Connect these to form a rectangle

See below:

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.