# Acute triangle

An acute triangle is a type of triangle in which the three internal angles have measures of less than 90°.

## What is an acute triangle

An acute triangle is characterized by having no angles that measure larger than 90°. Generally, triangles are categorized into 6 types: acute, obtuse, right, equilateral, isosceles, and scalene.

### Acute triangle definition

An acute triangle is a 3-sided polygon in which the interior angles (the angles formed inside the triangle) all measure less than 90°. An example of an acute triangle in real life is a yield sign:

Since all of the interior angles are 60°, the acute triangle above is more specifically described as an equilateral triangle. An acute triangle can be equilateral, isosceles, or scalene.

## Types of triangles

An acute triangle can be categorized as an equilateral triangle, isosceles acute triangle, or a scalene acute triangle.

### Equilateral triangle

An equilateral triangle is an acute triangle whose sides are all equal and whose interior angles all have the same measure. Since the interior angles of a triangle sum to 180°, the interior angles of an equilateral triangle are all equal to 60°. All equilateral triangles are acute triangles because their interior angles will always be less than 90° by definition. The figure below shows an example of an equilateral triangle.

In the equilateral triangle above, sides a = b = c and angles A = B = C.

### Isosceles acute triangle

An isosceles acute triangle is a triangle that has two sides of equal length and whose angles are all less than 90°. The figure below shows an isosceles acute triangle.

In the isosceles acute triangle above, sides a = b and angles A = B. Side c and angle C have different measures.

### Scalene acute triangle

A scalene acute triangle is a triangle whose sides all have different lengths and whose angles are all less than 90°. The figure below shows a scalene acute triangle.

In the scalene acute triangle above, none of the sides or angles are equal.

## Acute triangle properties

Below are some properties of acute triangles.

- The interior angles of an acute triangle are all between 0° and 90°.
- The points of concurrency (circumcenter, incenter, centroid, orthocenter) of an acute triangle all lie within the boundaries of the triangle.
- Given sides a, b, and c, where c is the longest side of an acute triangle: c
^{2}< a^{2}+ b^{2}. In other words, the square of the longest side of an acute triangle is always shorter than the sum of the squares of its shorter sides.

### Properties of triangles

Below are properties of all triangles.

- A triangle is a polygon with 3 sides, 3 angles, 3 vertices.
- The lengths of the sides of a triangle correspond to the measures of their angles; the larger the angle, the larger the side; the smaller the angle, the smaller the side. The longest side of a triangle is opposite the angle with the largest measure and the shortest side of the triangle is opposite the smallest angle.
- Angle sum property - The sum of the interior angles of a triangle is always equal to 180°. The sum of the exterior angles of a triangle is always equal to 360°
- Triangle inequality - The sum of the lengths of two sides of a triangle is always larger than the length of the third side; the difference between the lengths of any two sides is less than the length of the third side.
- Exterior angle theorem - The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles of the triangle.
- Congruent - Two triangles are congruent if all of their corresponding sides and angles are equal.
- Similar - Two triangles are similar if all their corresponding angles are equal and their corresponding sides have the same ratio. In other words, the shape of the triangle is identical, but the size of the triangles are different.
- The sum of consecutive interior and exterior angles of a triangle is supplementary (180°).

## Triangle formulas

Below are some formulas related to acute triangles.

### Perimeter of an acute triangle

The perimeter of an acute triangle is the sum of the sides of the triangle. Given a triangle with side lengths a, b, and c, the perimeter is:

P = a + b + c

If the lengths of the sides are not given, it may be possible to determine the side lengths of an acute triangle using trigonometry if enough sides or angles are known.

### Area of an acute triangle

The are a few different ways to find the area of an acute triangle. The typical triangle area formula is:

where b is the base of the triangle and h is the height. Any side of a triangle can be chosen as the base, and the height is the perpendicular line segment drawn from the vertex opposite the chosen base.

The figure above shows two orientations of the same triangle. In either case, the area of the triangle will be the same.

### Heron's formula

Heron's formula is used to find the area of a triangle if all 3 sides of the triangle are known. The area of a triangle using Heron's formula is

where a, b, and c are the sides of the triangle and s is the semiperimeter of the triangle:

### Converse of the Pythagorean theorem

When the lengths of the sides of a triangle are known, the converse of the Pythagorean Theorem can be used to determine whether or not the triangle is an acute triangle.

For a right triangle with a hypotenuse of length c and leg lengths a and b, the Pythagorean Theorem states:

a^{2} + b^{2} = c^{2}

On the other hand, in a triangle where a^{2} + b^{2} > c^{2}, if side c is also the longest side, the triangle is an acute triangle. Similarly, if a^{2} + b^{2} < c^{2}, the triangle is obtuse.

Example:

Is triangle ABC an acute triangle?

Side AB above is the longest side of triangle ABC. To check if ABC is an acute triangle, let c = 13, a = 10 and b = 9:

a^{2} + b^{2} = 10^{2} + 9^{2} = 181

c^{2} = 13^{2} = 169

181 > 169

Therefore triangle ABC is an acute triangle.