Chapters
Triangle Definition
Let's take a look at the most basic definition of a triangle.
Tri- | Prefix that means three |
Angle | The bed that forms between two lines |
Triangle | A polygon that has three sides, whose corners form three angles |
As you can see, the definition of triangle can be found within the word itself. No matter what kind of triangle you’re dealing with, a triangle will always have three sides and three angles.
Triangle Properties
Now that we know the definition of a triangle, let’s see some of the properties of triangles.
Property | Name | Description |
1 | Angle Sum Property | All 3 interior angles have a sum of 180 degrees |
2 | Length Sum Property | The sum of two sides is greater than the length of the last side |
3 | Largest Angle Property | The angle opposite the largest side is the largest angle |
4 | Exterior Angle Property | The exterior angle of any interior angle of the triangle is equal to the sum of the other two interior angles |
As you can see, these properties can get kind of tricky - what is an exterior angle and what is an ‘opposite’ side? Let’s take a look at these properties in depth.
Angle Sum Property
The angle sum property states that all three interior angles of a triangle have to add up to 180 degrees. Let’s take a look at what interior angles are.
Definition | |
Interior Angle | The interior angle of a triangle is any angle that is formed by the sides of the triangle and is inside the triangle |
Let’s take a look at some examples.
Can you tell which ones are exterior and interior triangles? Take a look at the table below and see if you answered correctly.
Interior? | |
A | No |
B | Yes |
C | Yes |
D | Yes |
E | No |
Length Sum Property
The length sum property states that the length of two sides is always greater than the last side. We can understand this property by picking the smallest length you can think of for all sides. Take a length of 1 cm for example.
AB + BC | 1+1 = 2 |
BC + CA | 1+1 = 2 |
CA + AB | 1+1 = 2 |
As you can see, the sum of any of the sides is always greater than the last side.
Largest Angle Property
The largest angle property states that the largest angle is opposite the largest side of a triangle. The proof for this property is quite complex, however it boils down to the fact that the larger the angle is, the longer of a line it projects.
A | B |
The angle of 80 degrees projects a longer line | The angle of 10 degrees projects a smaller line |
Exterior Angle Property
The exterior angle property states that the exterior angle of any interior angle of a triangle is equal to the sum of the other two interior angles. Let’s take a look at this in practice.
Exterior angle | 109 degrees |
Angle A + Angle C | 38 + 71 = 109 |
Equilateral Triangle
There are four different types of triangles. Let’s start by looking at an equilateral triangle. An equilateral triangle can be defined by the table below.
Equilateral Triangle | Definition |
1 | A triangle |
2 | Three equal sides |
3 | Three equal angles |
4 | Which are all 60 degrees |
It’s quite easy to identify the different types of triangles. Every triangle has a different notation. You should familiarize yourself with this notation so that you can understand what type of triangle you’re dealing with.
A | Three same angles |
B | Three same sides |
In order to find the perimeter or area of an equilateral triangle, simply follow the formulas below.
Perimeter | P = 3*s |
Area | A = |
Scalene Triangle
Out of the four triangles, let’s take a look at the next type of triangle: a scalene triangle. A scalene triangle can be defined by the table below.
Scalene Triangle | Definition |
1 | A triangle |
2 | No equal angles |
3 | No equal sides |
As you can see, a scalene triangle is quite special. This is the only type of triangle where at least two sides or angles are not equal. Remember that when two sides are the same length, they automatically form two angles of the same length. Let’s take a look at the notation for scalene triangles.
A | No angles are the same length, as market by the curves |
B | No sides are the same length, as marked by the lines |
In order to find the perimeter or area of a scalene triangle, simply follow the formulas below.
Perimeter | P = a+b+c |
Area | A = ½ *b*h |
Isosceles Triangle
The third most common triangle is the isosceles triangle. Let’s take a look at the properties of an isosceles triangle.
Isosceles Triangle | Definition |
1 | A triangle |
2 | Two equal angles |
3 | Two sides are the same |
As mentioned in the previous section, when two sides of a triangle are the same length, they form two angles that are the same size. Let’s take a look at the notation for this type of triangle.
Perimeter | P = 2*s+c |
Area | A = ½ *b*h |
Right Triangle
The last type of triangle is a right triangle. This type of triangle is the triangle with the most special properties. Let’s take a look at these properties.
Property | Description |
1 | Has one right angle (90 degrees) |
2 | The side opposite the right angle is the hypotenuse |
3 | The right angle is the largest angle of a right triangle |
A right triangle notation can be seen below.
Pythagorean Theorem
The Pythagorean theorem applies specifically to right triangles. It states that if you know that the length of the hypotenuse squared is equal to the sum of the squares of the other two sides. Let's take a look at what this actually means.
Formula | |
c | hypotenuse |
a | One side |
b | The other 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