Triangles are a basic and very familiar geometric shape. With three sides, the triangle is the simplest possible polygon (try imagining a two-dimensional solid with only two sides; you can get close, but not all the way there) and has a number of unique and interesting properties.

Some features are common to all triangles, just as every aircraft has to somehow produce enough lift to remain aloft. But triangles come in a number of distinct forms, some of which have properties unique to that class of triangle.

You have no doubt encountered isosceles triangles in your travels, but probably without recognizing that they had a special name and, along with this identity, certain special mathematical properties. Finding the area of an isosceles triangle is one of many straightforward exercises you can perform on this figure.

## Properties of Triangles

All triangles have three sides and three angles. Because this is the only restriction, the number of possible triangles is literally **infinite**. In practice, however, extremely tiny (that is, approaching 0 degrees) and extremely large (that is, approaching 180 degrees) angles are rarely encountered.

The sum of the angles in a triangle is always 180 degrees. If one of the three angles is 90 degrees (a right angle), the triangle is called a right triangle and can be quickly analyzed using trigonometric tools "regular" triangles cannot.

The area of any triangle is one-half its base times its height or:

Because of the shapes of certain triangles, it is not always easy to calculate the height even if you know the length of all three sides. Fortunately, this is not true of isosceles triangles.

## The Isosceles Triangle

An isosceles triangle is a triangle with two equal sides. Be very careful when you read that, because it does not say "*exactly two* equal sides." This means that a triangle with three equal sides, which by definition has three equal angles of 60 degrees each, is an isosceles triangle, but this one goes by a special name – equilateral triangle.

Isosceles triangles have the property of **bilateral symmetry**, meaning that they can be divided into two triangles of equal area that are mirror images of one another. When this is done, the result is two right triangles. These are not identical, but because their angles and sides have the same values, they are *congruent triangles*.

## Area of an Isosceles Triangle

If the height of the isosceles triangle is not given explicitly, but you are told the value of one of the sides and the base, you can calculate the height using basic trigonometry and proceed from there. If you know the height and one side, you can figure out the length of the base in a similar way and work toward the solution.

Regardless, the general form of the equation for the area of a triangle applies to an isosceles triangle:

## Isosceles Triangle Problem

Say you are visiting your grandfather, who has just bought a patch of land in the shape of a long, narrow isosceles triangle. He proudly tells you that he paid only $1,000 for it – $1 per square meter. You deduce that the plot is thus 1,000 m^{2} in area.

"The thing is," your grandfather tells you as you both stand at the "tip" of the patch of land looking toward the distant base, "I don't even know how wide it is down there. I just know it's 100 paces to get there, and each pace is exactly a meter, if memory serves."

You quickly pull out your calculator and tell your grandfather how wide the land patch is at its base. What is this value?**Answer:** If the area is 1,000 m^{2} and this is equal to (1/2)(b)(100 m) = (50 m)b, then b = 20 m. Also, if you are interested in the perimeter of the triangle, or the distance around its three sides, that is a problem you and your grandfather can take up independently!