Triangles are the simplest polygons you can meet in geometry; however, they are extremely interesting and hide an unexpected complexity: our AAA triangle calculator will help you with one of their fundamental properties.
What exactly is a triangle?
A triangle is a polygon with:
- Three sides, or edges; and
- Three angles, or vertices.
🙋 Triangles are the simplest bidimensional shapes. You can't build a polygon with only two sides; it would be a line!
Any set of three non-collinear points defines a unique triangle, and just to prove that triangles are cool, the same set also defines a unique circumference. This is true for every triangle, not just regular ones.
Triangles are summarily grouped in categories according to the relations between their sides or angles. Let's quickly see them.
For the sides, we have:
- A triangle with all the sides of equal length is an equilateral triangle;
- If only two sides are equal, we have an isosceles triangle; and
- If all sides are different, we are dealing with a scalene triangle.
Note that equilateral triangles are often considered to additionally be isosceles.
As for the angles, this is how we can distinguish triangles:
- If all interior angles are smaller than , the triangle is acute;
- If one of the angles is greater than , the triangle is obtuse;
- If one of the angles is equal to , we are dealing with the much-celebrated right triangles.
Let's focus on the angles!
The angles of a triangle
We are going to say it bluntly first, and then we will analyze it in detail:
In two dimensions, the sum of the interior angles of a triangle equals the straight angle.
We think this sentence is straightforward enough, but trust us, it is more interesting than it looks!
Let's call the angles of a triangle with the first three letters of the Greek alphabet: , , and . The statement above translates to:
This is a pretty strong constraint but an elegant one too. It implies, for example, that triangles can't be "twisted" by applying pressure on them. That's why triangular shapes appear everywhere, from powerline pylons to skyscrapers.
But how do we prove that this identity is true? We use the parallel postulate, which tells us that when we intersect two parallel lines with a third oblique line, the four angles we form are equal in pairs.
Take a triangle, and draw a line passing through a vertex and parallel to the opposite side. At the intersection, you can identify three angles.
- One of them is an interior angle of the triangle;
- The other two are the angles formed by the intersection between the two parallel lines and the remaining two sides of the triangle.
You can easily see that the three angles on the newly drawn line sum to , and now you know that all three of them have corresponding "clones" inside the triangle. Are you convinced?
This important relationship allows us to calculate the third angle of the set of interior angles with a simple subtraction.
Calculations on an AAA triangle
If you know the values of two interior angles, let's say and , you can calculate the third angle, , with the relations:
If you are measuring your angles in degrees, and:
If you are using radians.
That's it: the only calculations on an AAA triangle involves the angles!
Can we solve an AAA triangle?
Sadly, we can't solve an AAA triangle. Since the angles only define the general shape but not the scale of a polygon, we can't check for congruence in AAA triangles: unless a side is specified, we can't calculate an AAA triangle's sides.
In this case, we talk of similar triangles.
Other combinations of sides and angles in a triangle
Here at Omni Calculator, we love triangles. That's why we made a LOT of calculators about triangles. Check them out!
- Triangle area calculator;
- Midsegment of a triangle calculator;
- Acute triangle calculator;
- Circumcenter of a triangle calculator;
- Triangle congruence calculator;
- Obtuse triangle calculator;
- Oblique triangle calculator;
- Base of a triangle calculator;
- AAS triangle calculator;
- SAS triangle calculator;
- SSS triangle calculator; and
- ASA triangle calculator.
What does AAA triangle mean?
An AAA triangle is a triangle where only the three angles are defined. We can't solve an AAA triangle since we lack the scale of the polygon. However, if you know only two angles, you can easily complete it using the relation:
α + β + γ = 180°
What is the third angle of a triangle if α = 30° and β = 90°?
The last angle of a triangle with
α = 30° and
β = 90° is
γ = 60°. To find the result
- Apply the relation between the angles of a triangle:
α + β + γ = 180°
- Substitute the values of the known angles:
γ = 180° - α - β = 180° - 30° - 90° = 60 °
How do I calculate the third angle of a triangle?
To calculate the third angle of a triangle:
- Remember that the sum of the angles in a triangle is
- Isolate the unknown angle in the relation
α + β + γ = 180°;
- Substitute the values of the known angles in the equation.
Remember that if your angles are expressed in radians, their sum equals
Is there congruence in AAA triangles?
There is no congruence in AAA triangles. Since the angles define only the shape but not the scale, we don't have the data to say if two triangles with the same angles are congruent.
AAA triangles with the same set of values for
γ are called similar triangles.