## Types of Triangles

A triangle is a polygon that has three sides. From there, triangles are classified as either right triangles or oblique triangles. A right triangle has a 90° angle, while an oblique triangle has no 90° angle. Oblique triangles are broken into two types: acute triangles and obtuse triangles. Take a closer look at what these two types of triangles are, their properties, and formulas you'll use to work with them in math.

## Obtuse Triangles

## Obtuse Triangle Definition

An obtuse triangle is one that has an angle greater than 90°. Because all the angles in a triangle add up to 180°, the other two angles have to be acute (less than 90°). It's impossible for a triangle to have more than one obtuse angle.

## Properties of Obtuse Triangles

- The longest side of an obtuse triangle is the one opposite the obtuse angle vertex.
- An obtuse triangle may be either isosceles (two equal sides and two equal angles) or scalene (no equal sides or angles).
- An obtuse triangle has only one inscribed square. One of the sides of this square coincides with a part of the longest side of the triangle.
- The area of any triangle is 1/2 the base multiplied by its height. To find the height of an obtuse triangle, you need to draw a line outside of the triangle down to its base (as opposed to an acute triangle, where the line is inside the triangle or a right angle where the line is a side).

## Obtuse Triangle Formulas

To calculate the length of the sides:

c^{2}/2 < a^{2} + b^{2} < c^{2}

where angle C is obtuse and the length of the sides is a, b, and c.

If C is the greatest angle and h_{c} is the altitude from vertex C, then the following relation for altitude is true for an obtuse triangle:

1/h_{c}^{2} > 1/a^{2} + 1/b^{2}

For an obtuse triangle with angles A, B, and C:

cos^{2} A + cos^{2} B + cos^{2} C < 1

## Special Obtuse Triangles

- The Calabi triangle is the only non-equilateral triangle where the largest square fitting in the interior can be positioned in three different ways. It is obtuse and isosceles.
- The smallest perimeter triangle with integer length sides is obtuse, with sides 2, 3, and 4.

## Acute Triangles

## Acute Triangle Definition

An acute triangle is defined as a triangle in which all of the angles are less than 90°. In other words, all of the angles in an acute triangle are acute.

## Properties of Acute Triangles

- All equilateral triangles are acute triangles. An equilateral triangle has three sides of equal length and three equal angles of 60°.
- An acute triangle has three inscribed squares. Each square coincides with a part of a triangle side. The other two vertices of a square are on the two remaining sides of the acute triangle.
- Any triangle in which the Euler line is parallel to one side is an acute triangle.
- Acute triangles can be isosceles, equilateral, or scalene.
- The longest side of an acute triangle is opposite the largest angle.

## Acute Angle Formulas

In an acute triangle, the following is true for the length of the sides:

a^{2} + b^{2} > c^{2}, b^{2} + c^{2} > a^{2}, c^{2} + a^{2} > b^{2}

If C is the greatest angle and h_{c} is the altitude from vertex C, then the following relation for altitude is true for an acute triangle:

1/h_{c}^{2} < 1/a^{2} + 1/b^{2}

For an acute tirangle with angles A, B, and C:

cos^{2} A + cos^{2} B + cos^{2} C < 1

## Special Acute Triangles

- The Morley triangle is a special equilateral (and thus acute) triangle that is formed from any triangle where the vertices are the intersections of the adjacent angle trisectors.
- The golden triangle is an acute isosceles triangle where the ratio of twice the the side to the base side is the golden ratio. It is the only triangle that has angles in the proportion 1:1:2 and has angles of 36°, 72°, and 72°.