Chapters
Definition of a Triangle
Definition | |
Triangle | A triangle is a shape that is made up of 3 sides. Three interior angles are formed at the 3 points where these 3 sides meet. |
Take a look at an example of a standard triangle below.
Property 1 | Angle Sums | The sum of the interior angles is always 180 degrees. |
Property 2 | Triangle Inequality | The sum of lengths of two sides is always greater than the last side. |
Property 3 | Greatest angle | The side that is opposite the largest angle always has the longest length. |
Types of Triangles
There are many types of triangles. While they may have strange names, the name of any triangle can give you a hint to what type of triangle they are. Take a look at the name breakdown of triangle.
Tri- | Angle |
Means three | Comes from the Latin word angulus, which means corner |
As you can see, the word triangle actually tells you that you’re dealing with a shape that has three angles. Check out the three different types of triangle and pay attention to their names.
A | B | C |
Equilateral | Isosceles | Scalene |
As you can see, the three triangles have names that give you a hint as to what the properties of their sides and angles are. Take a look at the definitions below.
Angles | Sides | |
Equilateral | Three of the same, or ‘equal’, angles which are all 60 degrees | Three sides with the same, or ‘equal’, length |
Isosceles | Two equal angles | Two sides are the same |
Scalene | No equal angles | No equal sides |
You may have noticed that the triangles in the image above have special symbols. Let’s take a look at what these symbols mean:
A | B | C |
Equal angles | Equal sides | Two equal sides and one of a different length |
Types of Angles
Understanding angles is an important part of understanding many concepts related to triangles. We already talked about some of the properties that triangles have when it comes to their angles. Now, let’s take a closer look at different types of angles.
Acute angle | Right angle | Obtuse angle | |
Definition | Angle < 90 degrees | 90 degree angle | Angle > 90 degrees |
While this may seem strange, you can think of all angles in relation to right angles. If it’s between 0-89 degrees, it is an acute angle. If it’s between 91-180 degrees it is an obtuse angle.
A | B | C |
Acute angle | Right angle | Obtuse angle |
You should also note that the size of the angle determines the size of the length of the side. This makes sense because the side opposite the angle cannot be bigger or smaller than the sides next to the angle.
A | B |
The angle that the two sides produce determines how big the third side will be | Since the two angles are the same, the lengths of the two sides opposite the angles will also be the same |
Equilateral Triangle
As you saw in the previous section, an equilateral triangle is defined as a triangle with three equal angles and sides. Let’s take a closer look at the angles of an equilateral triangle.
Angle Sums | Equilateral Triangle Definition | Result |
All angles must = | 3 equal sides | 180 divided by 3 = |
As you can see, because the equilateral triangle has three equal angles, and because all angles must add up to 180, the angles will always be equal to 60 degrees regardless of the length.
Isosceles Triangle
Recall that the definition of an isosceles triangle is that there are two equal angles and two equal sides. Let’s take a look at what the word isosceles means.
Isos | Skelos |
Greek for “equal” | Greek for “leg” |
Here are the different kinds of isosceles triangles you can have.
Triangle 1 | Acute Isosceles | All angles less than 90 degree, two sides equal |
Triangle 2 | Right Isosceles | One angle 90 degrees, two sides equal |
Triangle 3 | Obtuse Isosceles | One angle greater than 90 degrees, two sides equal |
Scalene Triangle
A scalene triangle is defined as a triangle that has no equal sides and no equal angles. Here are some of the properties of scalene triangles.
Triangle 1 | Acute Isosceles | All angles less than 90 degree, no equal sides |
Triangle 2 | Right Isosceles | One angle 90 degrees, no equal sides |
Triangle 3 | Obtuse Isosceles | One angle greater than 90 degrees, no equal sides |
Perimeter of a Triangle
Finding the perimeter of a triangle depends on what type of triangle you are dealing with. Let’s take a look at the different formulas for the different types of triangles. Keep in mind that the notation for any perimeter is written as .
Formula | |
Scalene Triangle | P = a + b + c |
Formula | |
Isosceles Triangle | P = 2*s + b |
Formula | |
Equilateral Triangle | P = 2s |
Area of a Triangle
The area formula for a triangle is a bit easier, as there is only one area formula. Take a look at the formula below.
Formula | |
Any Triangle |
As you can see, while it might be easier to calculate the area, it can be difficult to actually solve for the area if you don’t have the size of the base or the height.
Pythagorean Theorem
The Pythagorean Theorem is one of the most important theorems in math. It only applies to right triangles, however, so be careful when you use it. The Pythagorean theorem is written below.
Definition | Formula | |
Pythagorean Theorem | The sum of the square of the two sides adjacent to the right angle is equal to the square of the longest side. |
I’m just curious if the area between the polygon and the circumscribed circle has a name.
https://www.superprof.co.uk/resources/academic/maths/geometry/plane/orthocenter-centroid-circumcenter-and-incenter-of-a-triangle.html