# Types of Triangles & Their Properties

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Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson introduces different types of triangles. We'll discuss some different characteristics that are common to all triangles and some properties that are unique to certain types of triangles.

## The Triangle Types

You are probably familiar with a triangle as a polygon with three sides and three angles. For instance, if you were to be shown a picture like this one with two shapes on it, each having three straight sides, you'd most likely be able to identify these shapes as triangles.

However, what you may not be aware of is that there are a number of different types of triangles. Let's look at them now:

• First there's the equilateral triangle. This is a triangle with all sides having equal lengths, and all angles having equal measures of 60 degrees. Triangle A is an equilateral triangle.

• Next is a right triangle. This is a triangle with one right angle (or an angle that measures 90 degrees) and two acute angles, where an acute angle is an angle that measures less than 90 degrees. Triangle B is a right triangle.

• Then there's the isosceles riangle This is a triangle with two sides of equal lengths.

• After that is a scalene triangle. This is a triangle with all three sides having different lengths.

• Next, there's the acute triangle, which is a triangle with three acute angles.

• Finally, there's the obtuse triangle. This is a triangle with two acute angles and one obtuse angle, where an obtuse angle is an angle that measures greater than 90 degrees.

There are also many properties that these different types of triangles satisfy. It would be impossible to list them all in one lesson, so we'll just concentrate on some of the important ones all triangles have in common, like base, altitude, height, and area.

## Base, Altitude, Height, and Area

The base of a triangle refers to the bottom side of the triangle. Any side can be a base when it is considered to be the bottom side of the triangle. The altitude of a triangle is a line that is perpendicular to the base of a triangle and passes through the corner opposite the base. The length of the altitude, from the base to the opposite corner, is the height of the triangle. Lastly, the area is the amount of space inside the triangle.

These four parts of a triangle all come together in the formula for the area of a triangle, which is:

A = (1/2)bh

where b = base length and h = height (or altitude length)

For example, if a triangle has base length 4 centimeters, and altitude length 10 centimeters, then we can find the area of the triangle by plugging b = 4 and h = 10 into the formula and simplifying.

A = (1/2)(4)(10) = 20

We see that the area of the triangle described is 20 square centimeters.

When it comes to the altitude of a triangle, there is one type of triangle where the altitude can be one of the sides, and that is the right triangle. A right triangle has a 90 degree angle, so two of its sides are perpendicular. If we consider one of the perpendicular sides to be the base, then the other perpendicular side is perpendicular to the base and passes through the corner opposite the base, making it the altitude.

Another triangle with a special altitude is the equilateral triangle. The altitude of an equilateral triangle splits the base exactly in half, and splits the triangle into two equal right triangles

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