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Area vs Perimeter of a triangle
 
A triangle with a fixed perimeter can have many different areas.
Try this Drag the orange dot on the triangle below. The triangle will have a fixed perimeter, but the area will vary.

A common error is to assume that a triangle that has a fixed perimeter must also have a fixed area. This is definitely not the case as can be seen from the figure above. As you drag the orange point A, the triangle will maintain a fixed perimeter. But as you can see, the area varies quite a bit.

When A is half way between B and C, the area is at a maximum. As you drag it around towards one side you can see the area decreasing, both in the formula at the top and by noticing that fewer and fewer squares can fit inside it. Eventually, when A is in line with B and C, the area is zero.

The area is at a maximum when the triangle is isosceles. That is, when both sides have the same length. Carefully adjust A above to create an isosceles triangle and note the area is the greatest when AC and AB are both the same length (9.0)

Try it with string

Make a loop of string and pass it around two pins (corresponding to the two points B and C above). Pull the string taut with a third pin to make a triangle. As you move any pin with the string tight, you will be making triangles with different areas but the perimeter is fixed (the length of the string loop).

The Ellipse Connection

In the figure above, select the "Show trail" checkbox, then drag point A all the way around the base line. The resulting shape is an ellipse.

Why is this? The definition of an ellipse is

"A line forming a closed loop, where the sum of the distances from two points (foci) to every point on the line is constant"
Points B and C form the two foci. Since the distance from B to C is fixed, and the perimeter is fixed, then the sum of the distances AB and AC are constant - the condition required to form an ellipse.

Learn more about Ellipses

The string experiment described above is actually a practical way to draw an ellipse. See Drawing an ellipse using string and 2 pins. For more on ellipses, see also Definition of an ellipse.

Other triangle topics

General

Perimeter / Area

Triangle types

Triangle centers

Congruence and Similarity

Solving triangles

Triangle quizzes and exercises