The obtuse triangle is one where one of its interior angles is obtuse, that is, greater than 90º. Also, the other two angles are acute, which means they measure less than 90º.
This type of triangle is a very particular case within the types of triangle according to the measure of their internal angles.
It should be noted that the triangle is a polygon that cannot have more than one obtuse interior angle because its three interior angles must add up to 180º. So if one measures 91, for example, the other two must add up to 89º.
At this point, it is worth remembering that a polygon is a two-dimensional geometric figure that is made up of the union of different points (that are not part of the same line) by line segments. In this way, a closed space is built.
Another issue to mention is that the obtuse triangle is a type of oblique triangle that is one that does not have a right interior angle (which measures 90º).
Elements of the obtuse triangle
Guiding us from the figure below, the elements of the obtuse triangle are the following:
- Vertices: A, B, C.
- Sides: AB, BC, AC.
- Interior angles: ∝, β, γ. They all add up to 180º.
- Exterior angles: e, d, h. Each is supplementary to the interior angle of the same vertex. That is, it is true that: 180º = ∝ + d = β + e = h + γ. This implies that two of the exterior angles are obtuse and one is acute (the one that corresponds to the obtuse interior angle). If β measures 92º, for example, e would measure 88º.
Types of obtuse triangle
The types of obtuse triangle, according to the measure of its sides, are the following:
- Isosceles: Two of its sides measure the same and the other is different.
- Scalene: All its sides and interior angles are different.
Perimeter and area of the obtuse triangle
The characteristics of the obtuse triangle can be measured based on the following formulas:
- Perimeter (P): It is the sum of the sides that, observing the figure above where we indicate the elements, would be: P = a + b + c.
- Area (A): In this case, we are based on Heron's formula where s is the semiperimeter, that is, P / 2.
Example of obtuse triangle
Suppose a triangle has two interior angles that measure 40º and 45º degrees. Is it an obtuse triangle?
If all the interior angles add up to 180º, we can find the third unknown angle (x):
180º = 40º + 45º + x
180º = 85º + x
x = 95º
Since x is more than 90º, it is an obtuse angle. Therefore, we are faced with an obtuse triangle.
Now let's look at another exercise. Let's look at the following figure:
Suppose side BC (a) is 25 meters. α measures 35º, and β measures 45º. What is the perimeter and area of the figure?
First, we'll build on the sine theorem, dividing the length of each side by the sine of its opposite angle:
Also, if α + β + γ = 180, then:
35 + 45 + γ = 180
80 + γ =180
γ = 100º
Therefore, it is an obtuse triangle case.
We solve for b:
We solve for c:
Then, we calculate the perimeter and the semi-perimeter with the formula presented previously:
P = 25 + 30.8201 + 42.92240 = 98.7441 meters
S = P / 2 = 49.3720
Finally, we calculate the area with the formula presented previously