Scalene Triangle

A scalene triangle is a triangle in which all three sides have different lengths and all angles are diferent too.

This type of triangle can be a right triangle (one angle with measure 90°) but not all scalene triangles have a right angle. The scalene triangles can be acute (all interior angles measuring less than 90°), obtuse (one interior angle measuring more than 90°) or right triangles depending on their angles.

These three interior angles add to 180° (α+β+γ=180º) as in all of triangles.

Why do the angles of a triangle add up to 180°?

Area of a Scalene Triangle

The area of a scalene triangle can be calculated using Heron’s formula if all its sides (a, b and c) are known.

It could also be calculated if a side (b) is known and the height (h) associated with that side.

Finally, the area of a scalene triangle can also be calculated by trigonometric procedures, provided that three elements of the triangle are known, at least one of which is a side.

Perimeter of a Scalene Triangle

A scalene triangle is a triangle in which all three sides are in different lengths. The perimeter of a scalene triangle with three unequal sides is determined by adding the three sides.

If we know the length of the three sides you can calculate its perimeter using the following formula:

If only the length of two sides and the angle between them are given, in order to find the perimeter we need to calculate the other side using the Law of Cosines (Cosine Rule):

Solved Exercises

Exercise of the Area of a Scalene Triangle

Find the area of the scalene triangle given its three sides: a=2 cm, b=4 cm and c=3 cm.

What is its area?

We can calculate the area using Heron’s formula. First, we have to determine the semiperimeter s:

Now, we can apply the Heron’s formula:

So, the area is 2.9 cm².

Exercise of the Perimeter of a Scalene Triangle

Consider a given triangle:

Let be a scalene triangle of known sides, these being a=2 cm, b=4 cm y c=3 cm.

What is its perimeter? This will be calculated as the sum of its three sides.

So, the perimeter is 9 cm.

Exercise 3

What is its perimeter?

Solution:

In this triangle we only know two sides (a=4 cm y c=6 cm) and the angle they form (B=85°). First we have to find the length of side b.

For this we use the Law of Cosines:

So, we get the side b=6.92 cm.

Now we can calculate the perimeter by adding the three sides of the triangle:

The perimeter of the triangle is 16.92 cm.