# Equilateral, Isosceles and Scalene Triangles

A triangle is a three-sided polygon that has three vertices and three interior angles. The sum of the interior angles in a triangle is always equal to 180 degrees. If we take into account the length of their sides, we can distinguish three types of triangles: equilateral triangle, isosceles triangle, and scalene triangle. The equilateral triangle is characterized by having all its sides with equal length. The isosceles triangle has two equal sides and one unequal side. The scalene triangle has all its sides with different lengths.

Here, we will learn about the equilateral, isosceles, and scalene triangle in more detail, using diagrams to illustrate the concepts.

##### GEOMETRY

Relevant for

Learning about equilateral, isosceles, and scalene triangles.

See triangles

##### GEOMETRY

Relevant for

Learning about equilateral, isosceles, and scalene triangles.

See triangles

## Equilateral triangles

An equilateral triangle has the main characteristic that all its sides are equal. Furthermore, all its internal angles are also equal. We know that a triangle has a sum of 180° in its interior angles, so each angle in an equilateral triangle measures 60°.

This triangle can be considered as a special case of the isosceles triangle, where the third side also has the same length. The triangle ABC that we have next has the sides AB = BC = CA.

### Properties of an equilateral triangle

The main properties of these triangles are:

• They have 3 equal sides.
• They have 3 equal angles.
• Each interior angle of an equilateral triangle measures 60°.
• This triangle is a regular polygon with 3 sides.
• The height and the mean of a vertex represent the same line.
• The orthocenter and the centroid of these triangles are the same point.

## Isosceles triangles

An isosceles triangle is a triangle that has two equal sides, regardless of the direction of the apex of the triangle’s points. Since they have two equal sides, they also have two equal angles. In the triangle ABC below, we have AB = AC. Similarly, in the triangle PQR, we have PQ = PR.

The vertices A and P are known as the peaks or apices and the sides BC and QR, which are the unequal sides, are known as the bases of the isosceles triangle.

### Properties of an isosceles triangle

Some of the fundamental properties of these triangles are:

• These triangles have two equal sides.
• They have two equal angles called the base angles.
• When the third angle is 90 degrees, we have an isosceles right triangle.
• The height from the apex to the base bisects the angle at the apex.

## Scalene triangles

A scalene triangle is characterized by having three sides with different lengths. Also, the three angles in a scalene triangle also have different measures. In the scalene triangle ABC shown below, we have AB ≠ BC ≠ CA.

### Properties of a scalene triangle

The following are some of the most important properties of a scalene triangle:

• All its sides are uneven.
• All of its angles are unequal.
• The sum of the three angles is equal to 180 degrees.

## Important triangle formulas

The most important formulas for triangles are the area formula and the perimeter formula.

### Area of a triangle

The area of any triangle can be calculated using the length of its height and the length of its base. For this, we use the formula:

where b is the length of the base and h is the length of the height.

### EXAMPLE

A triangle has a base of 10 m and a height of 8 m. What is its area?

Solution: We use the formula for the area of a triangle with lengths $latex b=10$ and $latex h=8$:

$latex A=\frac{1}{2}\times b \times h$

$latex A=\frac{1}{2}\times 10 \times 8$

$latex A=40$

The area of the triangle is 40 m².

### Perimeter of a triangle

The perimeter of any geometric figure is equal to the sum of the lengths of its sides. In the case of triangles, we have:

where, $latex a, ~b, ~c$ are the lengths of the sides.

If we have an equilateral triangle, we know that we have three equal sides, so the perimeter formula becomes $latex p=3a$, where, a represents the length of one of the sides.

If we have an isosceles triangle, we know that we have two equal sides, so we can calculate the perimeter using the formula $latex p=2a+b$, where, a represents the length of the equal sides and b represents the length of the base (the uneven side).

### EXAMPLE

What is the perimeter of an isosceles triangle that has equal sides of length 12 m and a base of length 13 m?

Solution: We have an isosceles triangle and we have the values $latex a=12$ and $latex b=13$. Therefore, we plug these values into the formula the perimeter:

$latex p=2a+b$

$latex p=2(12)+13$

$latex p=24+13$

$latex p=37$

The perimeter of the triangle is 37 m.